Not sure why there weren't more takers for this quiz...maybe there have been just too many positions posted here this week. Anyway, here are the rollouts and analyses that I promised.
Problem 1
When this position came up OTB, I played 7/5 and looked around for a 3. But 24/22 is mandatory here, and then the only plausible 3 is 8/5, making the 5pt. I've found it to be a surprisingly good rule of thumb that if your opponent has an advanced anchor while you still have a 24pt anchor, then you must try to split off that anchor. Here, White's strong board would have made me balk at playing 24/22 8/5 and leaving a double shot even if I had seen that play, but actually, White's strong board makes it even more urgent to get a better anchor.
In the variant position, I've changed the roll to 42, and anchoring on the 20pt comes out on top by a sizable margin (according to XGR+) despite many seemingly plausible alternatives.
  White is Player 1
score: 0 pip: 136  Money session Jacoby Beaver  pip: 174 score: 0
Blue is Player 2  
XGID=bADABcDabcbAbB:0:0:1:32:0:0:3:0:10 
Blue to play 32 
1.  Rollout^{1}  24/22 8/5  eq: 0.526 
 Player: Opponent:  40.21% (G:9.01% B:0.45%) 59.79% (G:27.30% B:1.70%)  Conf: ± 0.026 (0.552...0.500) Duration: 2 hours 26 minutes 

2.  Rollout^{1}  7/5 6/3  eq: 0.640 (0.113) 
 Player: Opponent:  38.12% (G:8.54% B:0.45%) 61.88% (G:27.08% B:2.92%)  Conf: ± 0.023 (0.663...0.617) Duration: 1 hour 53 minutes 

3.  Rollout^{1}  13/10 7/5  eq: 0.662 (0.135) 
 Player: Opponent:  37.50% (G:7.82% B:0.38%) 62.50% (G:29.23% B:3.41%)  Conf: ± 0.030 (0.692...0.632) Duration: 2 hours 35 minutes 

4.  Rollout^{1}  24/22 6/3  eq: 0.689 (0.163) 
 Player: Opponent:  38.18% (G:7.49% B:0.41%) 61.82% (G:32.69% B:2.22%)  Conf: ± 0.034 (0.723...0.655) Duration: 3 hours 24 minutes 

5.  Rollout^{1}  24/22 5/2  eq: 0.737 (0.210) 
 Player: Opponent:  34.52% (G:6.68% B:0.26%) 65.48% (G:25.51% B:1.51%)  Conf: ± 0.025 (0.762...0.712) Duration: 2 hours 27 minutes 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21
Variant:
  White is Player 1
score: 0 pip: 136  Money session Jacoby Beaver  pip: 174 score: 0
Blue is Player 2  
XGID=bADABcDabcbAbB:0:0:1:42:0:0:3:0:10 
Blue to play 42 
1.  XG Roller+  24/20 22/20  eq: 0.544 
 Player: Opponent:  40.33% (G:8.55% B:0.38%) 59.67% (G:26.38% B:1.39%)  

2.  XG Roller+  22/18 7/5  eq: 0.610 (0.067) 
 Player: Opponent:  38.50% (G:8.42% B:0.37%) 61.50% (G:24.63% B:2.03%)  

3.  XG Roller+  24/22 7/3  eq: 0.617 (0.073) 
 Player: Opponent:  38.00% (G:7.97% B:0.35%) 62.00% (G:24.51% B:1.39%)  

4.  XG Roller+  13/9 7/5  eq: 0.676 (0.132) 
 Player: Opponent:  37.83% (G:8.10% B:0.34%) 62.17% (G:27.20% B:2.97%)  

5.  XG Roller+  24/22 24/20  eq: 0.712 (0.169) 
 Player: Opponent:  38.21% (G:7.33% B:0.31%) 61.79% (G:29.18% B:1.65%)  

6.  XG Roller+  7/5 6/2  eq: 0.735 (0.191) 
 Player: Opponent:  35.50% (G:8.05% B:0.32%) 64.50% (G:25.71% B:2.40%)  

7.  XG Roller+  24/20 7/5  eq: 0.788 (0.244) 
 Player: Opponent:  36.04% (G:8.14% B:0.35%) 63.96% (G:31.02% B:2.15%)  

8.  XG Roller+  7/3 5/3  eq: 0.839 (0.295) 
 Player: Opponent:  33.88% (G:7.04% B:0.26%) 66.12% (G:26.00% B:2.23%)  

9.  XG Roller+  24/22 6/2  eq: 0.843 (0.300) 
 Player: Opponent:  35.93% (G:7.21% B:0.31%) 64.07% (G:29.89% B:2.16%)  

10.  XG Roller+  24/22 13/9  eq: 0.867 (0.324) 
 Player: Opponent:  37.86% (G:7.09% B:0.34%) 62.14% (G:34.58% B:3.35%)  

11.  XG Roller+  13/7  eq: 0.869 (0.326) 
 Player: Opponent:  35.91% (G:6.97% B:0.30%) 64.09% (G:30.23% B:3.60%)  

12.  XG Roller+  24/22 5/1  eq: 0.883 (0.339) 
 Player: Opponent:  32.32% (G:6.33% B:0.21%) 67.68% (G:24.10% B:1.44%)  

13.  XG Roller+  22/16*  eq: 0.927 (0.384) 
 Player: Opponent:  36.83% (G:6.99% B:0.30%) 63.17% (G:32.55% B:4.78%)  

eXtreme Gammon Version: 1.21
Problem 2
Pay now or pay later? Answer: Both!
Though White's board will get stronger by the minute, leaving a double direct shot by making the 9pt right away rolls out as a double whopper. Given that, 13/4 would seem to be the natural move, but the rollout favors the unusual move 13/9 7/2 by a clear margin.
BGO readers with excellent memories may realize that the way I came up with this problem was to tinker with this old position that I stole from Grunty on rec.games.backgammon. In that position, 13/9 7/2 came out on top but by only a small amount; also, paying now with 14/9 13/9 wasn't all that far behind. I said that I though very few humans would find that move, and Stick replied, "I believe less humans would make that play than a few." But Sharikov managed to find it, perhaps with the benefit of QF. I confess that I still don't quite understand the play. It's true that White has to break anchor to hit and has no board to contain the hit checker, but does Blue really gain that much from the 9pt blot as a landing spot, as Sharikov suggests?
In the variant position, I've strengthened White's board.
  White is Player 2
score: 0 pip: 135  4 point match  pip: 95 score: 0
Blue is Player 1  
XGID=BAAABBCbcABceb:1:1:1:54:0:0:0:4:10 
Blue to play 54 
1.  Rollout^{1}  13/9 7/2  eq: +0.319 
 Player: Opponent:  69.52% (G:9.51% B:0.51%) 30.48% (G:3.88% B:0.31%)  Conf: ± 0.005 (+0.314...+0.324) Duration: 6 hours 56 minutes 

2.  Rollout^{1}  13/4  eq: +0.257 (0.062) 
 Player: Opponent:  68.95% (G:5.67% B:0.31%) 31.05% (G:3.25% B:0.16%)  Conf: ± 0.005 (+0.252...+0.262) Duration: 5 hours 01 minute 

3.  Rollout^{1}  7/3 7/2  eq: +0.254 (0.065) 
 Player: Opponent:  67.08% (G:9.40% B:0.38%) 32.92% (G:3.78% B:0.23%)  Conf: ± 0.005 (+0.249...+0.259) Duration: 4 hours 36 minutes 

4.  Rollout^{2}  14/9 13/9  eq: +0.124 (0.196) 
 Player: Opponent:  64.00% (G:10.94% B:0.60%) 36.00% (G:4.44% B:0.43%)  Conf: ± 0.009 (+0.115...+0.133) Duration: 2 hours 35 minutes 


^{1} 7776 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply
^{2} 2592 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21, MET: RockwellKazaross
Variant:
  White is Player 2
score: 0 pip: 127  4 point match  pip: 95 score: 0
Blue is Player 1  
XGID=BAAABBCbcABccbb:1:1:1:54:0:0:0:4:10 
Blue to play 54 
1.  Rollout^{1}  13/4  eq: +0.260 
 Player: Opponent:  69.06% (G:3.94% B:0.40%) 30.94% (G:3.00% B:0.22%)  Conf: ± 0.010 (+0.250...+0.270) Duration: 1 hour 01 minute 

2.  Rollout^{1}  13/9 7/2  eq: +0.248 (0.012) 
 Player: Opponent:  67.14% (G:7.71% B:0.50%) 32.86% (G:4.30% B:0.36%)  Conf: ± 0.011 (+0.237...+0.259) Duration: 47 minutes 34 seconds 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21, MET: RockwellKazaross
Problem 3
At first glance, this one looks like a position that one might be able to calculate. A hit and a dance is a D/P, for 44% wins; if we toss in a few freak wins after getting hit back, we get winning chances in the mid to high 40's after 20/12*. If we don't hit, then there are two plausible plans: closing the board, or stepping out with 20/18(2) and halfclosing the board with 8/6. The latter play (my choice OTB) generates more shots because it threatens both blots directly, but since it doesn't close the board, the shots are not quite as deadly. Estimating the winning chances in these two scenarios looks feasible (though timeconsuming to do over the board); however, the real question is, how many gammons and backgammons do we lose after 20/12*?
It's clear we lose a ton of gammons but I'll be darned if I could estimate them OTB. In this particular position you might arrive at 20/12* anyway even with a notsogood estimate, because the difference in equity is so large. However, if we change the score to 5a5a (see variant), thereby elevating the gammon value, the plays get closer and arriving at the right play via calculation seems very difficult. By the way, eXtreme Gammon version 1.21 doesn't even register 20/12* as one of its top 32 plays at 5a5a, and I had to do the old "Setup > Play from position" trick to force it to analyze the move.
  White is Player 2
score: 0 pip: 63  Money session Jacoby Beaver  pip: 99 score: 0
Blue is Player 1  
XGID=ABBBCAAAaacBbcbc:1:1:1:22:0:0:3:0:10 
Blue to play 22 
1.  Rollout^{1}  20/12*  eq: 0.357 
 Player: Opponent:  43.01% (G:8.40% B:0.21%) 56.99% (G:41.45% B:3.29%)  Conf: ± 0.007 (0.364...0.350) Duration: 13 minutes 04 seconds 

2.  Rollout^{1}  11/9 8/6 5/1  eq: 0.498 (0.141) 
 Player: Opponent:  24.02% (G:0.49% B:0.01%) 75.98% (G:0.83% B:0.00%)  Conf: ± 0.003 (0.501...0.495) Duration: 21 minutes 51 seconds 

3.  Rollout^{1}  20/18(2) 11/9 8/6  eq: 0.504 (0.147) 
 Player: Opponent:  22.48% (G:1.11% B:0.02%) 77.52% (G:1.25% B:0.03%)  Conf: ± 0.003 (0.507...0.501) Duration: 10 minutes 13 seconds 

4.  Rollout^{1}  20/18(2) 8/6 5/3  eq: 0.507 (0.149) 
 Player: Opponent:  22.21% (G:1.03% B:0.02%) 77.79% (G:1.43% B:0.03%)  Conf: ± 0.003 (0.510...0.504) Duration: 8 minutes 32 seconds 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21
Variant:
  White is eXtremeGammon
score: 0 pip: 63  5 point match  pip: 99 score: 0
Blue is Player 1  
XGID=ABBBCAAAaacBbcbc:1:1:1:22:0:0:0:5:10 
Blue to play 22 
1.  Rollout^{1}  20/12*  eq: 0.470 
 Player: Opponent:  43.01% (G:8.42% B:0.28%) 56.99% (G:41.35% B:3.18%)  Conf: ± 0.008 (0.478...0.462) Duration: 12 minutes 50 seconds 

2.  Rollout^{1}  11/9 8/6 5/1  eq: 0.496 (0.026) 
 Player: Opponent:  23.98% (G:0.52% B:0.01%) 76.02% (G:0.83% B:0.00%)  Conf: ± 0.003 (0.499...0.493) Duration: 22 minutes 54 seconds 

3.  Rollout^{1}  20/18(2) 11/9 8/6  eq: 0.501 (0.031) 
 Player: Opponent:  22.44% (G:1.10% B:0.02%) 77.56% (G:1.24% B:0.03%)  Conf: ± 0.003 (0.504...0.498) Duration: 10 minutes 21 seconds 

4.  Rollout^{1}  20/18(2) 8/6 5/3  eq: 0.505 (0.035) 
 Player: Opponent:  22.16% (G:0.99% B:0.03%) 77.84% (G:1.40% B:0.03%)  Conf: ± 0.003 (0.508...0.502) Duration: 8 minutes 27 seconds 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21, MET: RockwellKazaross
Problem 4
Again, if you have a good memory, you might recognize this as a variant of this position I posted about a year ago. That was from a money game; here it's DMP, so even if you remembered the other position you might decide that 6/3* is better here, since we're not afraid of losing a gammon. However, the rollout favors the ugly but safe 8/6 2/1.
In the variant, I've weakened White's board slightly and the rollout is a dead heat.
  White is Player 2
score: 0 pip: 113  1 point match  pip: 106 score: 0
Blue is Player 1  
XGID=BBaDbCDaabbbbb:0:0:1:12:0:0:0:1:10 
Blue to play 12 
1.  Rollout^{1}  8/6 2/1  eq: 0.049 
 Player: Opponent:  47.55% (G:3.86% B:0.02%) 52.45% (G:10.86% B:0.17%)  Conf: ± 0.005 (0.054...0.044) Duration: 52 minutes 58 seconds 

2.  Rollout^{1}  6/3*  eq: 0.109 (0.060) 
 Player: Opponent:  44.55% (G:4.09% B:0.04%) 55.45% (G:17.91% B:0.46%)  Conf: ± 0.005 (0.114...0.104) Duration: 1 hour 04 minutes 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21, MET: RockwellKazaross
Variant:
  White is Player 2
score: 0 pip: 111  1 point match  pip: 106 score: 0
Blue is Player 1  
XGID=BBaDbCDaabbbbb:0:0:1:12:0:0:0:1:10 
Blue to play 12 
1.  Rollout^{1}  8/6 2/1  eq: 0.045 
 Player: Opponent:  47.74% (G:4.32% B:0.04%) 52.26% (G:10.66% B:0.14%)  Conf: ± 0.005 (0.050...0.040) Duration: 49 minutes 33 seconds 

2.  Rollout^{1}  6/3*  eq: 0.045 
 Player: Opponent:  47.74% (G:4.94% B:0.07%) 52.26% (G:17.23% B:0.24%)  Conf: ± 0.005 (0.050...0.040) Duration: 57 minutes 00 second 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21, MET: RockwellKazaross
Problem 5
Slot boldly or play safe? If we slot then it's fairly clear that 8/3 7/4 is best since it diversifies our numbers to cover and escape. This diversification is important; see the variant position below, and compare the absolute equity of 8/3 7/4 (+0.332±0.011) in the variant position with the absolute equity of 8/5 7/2 (+0.348±0.010) in the original position. It's clear we gain a lot if the slot works but we also lose big if it doesn't. I played safe here and the rollout says it was a whopper.
  White is Player 1
score: 0 pip: 118  Money session Jacoby Beaver  pip: 103 score: 0
Blue is Player 2  
XGID=aBBBBBABbacbbbBb:1:1:1:53:0:0:3:0:10 
Blue to play 53 
1.  Rollout^{1}  8/3 7/4  eq: +0.461 
 Player: Opponent:  62.43% (G:43.48% B:0.34%) 37.57% (G:8.83% B:0.38%)  Conf: ± 0.009 (+0.452...+0.470) Duration: 30 minutes 04 seconds 

2.  Rollout^{1}  8/3 6/3  eq: +0.423 (0.038) 
 Player: Opponent:  61.51% (G:42.01% B:0.27%) 38.49% (G:10.85% B:0.47%)  Conf: ± 0.009 (+0.414...+0.432) Duration: 25 minutes 45 seconds 

3.  Rollout^{1}  8/5 8/3  eq: +0.406 (0.055) 
 Player: Opponent:  60.86% (G:41.60% B:0.33%) 39.14% (G:8.49% B:0.39%)  Conf: ± 0.010 (+0.396...+0.416) Duration: 20 minutes 06 seconds 

4.  Rollout^{1}  8/5 7/2  eq: +0.348 (0.114) 
 Player: Opponent:  61.40% (G:35.26% B:0.24%) 38.60% (G:7.79% B:0.34%)  Conf: ± 0.010 (+0.338...+0.358) Duration: 25 minutes 39 seconds 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21
Variant:
  White is Player 1
score: 0 pip: 118  Money session Jacoby Beaver  pip: 102 score: 0
Blue is Player 2  
XGID=aBBBBBBAbacbbbBb:1:1:1:53:0:0:3:0:10 
Blue to play 53 
1.  Rollout^{1}  8/3 7/4  eq: +0.332 
 Player: Opponent:  59.19% (G:39.19% B:0.28%) 40.81% (G:9.14% B:0.45%)  Conf: ± 0.011 (+0.321...+0.343) Duration: 25 minutes 34 seconds 

2.  Rollout^{1}  7/4 7/2  eq: +0.331 (0.002) 
 Player: Opponent:  60.89% (G:34.92% B:0.24%) 39.11% (G:7.88% B:0.33%)  Conf: ± 0.011 (+0.320...+0.342) Duration: 32 minutes 03 seconds 

3.  Rollout^{1}  8/3 6/3  eq: +0.327 (0.006) 
 Player: Opponent:  59.34% (G:36.65% B:0.21%) 40.66% (G:6.57% B:0.22%)  Conf: ± 0.010 (+0.317...+0.337) Duration: 21 minutes 10 seconds 

4.  Rollout^{1}  8/5 7/2  eq: +0.284 (0.049) 
 Player: Opponent:  59.61% (G:33.36% B:0.25%) 40.39% (G:8.13% B:0.29%)  Conf: ± 0.011 (+0.273...+0.295) Duration: 31 minutes 13 seconds 


^{1} 1296 Games rolled with Variance Reduction. Dice Seed: 2 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21