For the first part:
If test is positive, we are 99% sure of correctness
If test is negative, we are 99% sure of correctness
99% of athletes DO NOT take the drug. Which means 1% do.
1% of 99% who have not actually taken the drug will test positive (false positive).
99% of 1% who have taken the drug will test positive (true positive).
Total positive tests = false positives + true positives = $0.01 \times 0.99 + 0.99 \times 0.01 = 0.0198 = 1.98%$ So 1.98% of tests are positive.
Of these, 0.99% are false positives and 0.99% are true positives, so the probability an athlete has taken the drug given that they tested positive is 50%.
If both athletes are drug tested:
If both athletes take the drug, then each has 1%*(100%1%*50%)=0.95% chance of winning if he luckily passes the test, unless the other athlete also passes and beats him.
If neither takes the drug, then each has 99%*(99%*50% + 1%)= 49.995% chance of winning by passing the drug test and beating the opponent, or if the opponent fails the drug test.
If one athlete takes the drug and the other doesn't, then the drugtaking athlete has 1%*(99%*75% + 1%) = 0.7525% chance of winning by passing the drug test and beating the opponent, or if the opponent fails the drug test. Similarly, the undoped athlete has 99%*(1%*25%+99%) = 98.2575% chance of winning. So the payoff matrix is now
Dave/Joe (%)  Drug  No Drug 
Drug  0.95/0.95 
0.7525/98.2575

No Drug 
98.2575/0.7525

49.995/49.995

Dave/Joe (%)  Drug  No Drug 
Drug  0.5/99.5  0.75/99.25 
No Drug  24.75/75.25  49.5/50.5 
A/B  Drug  No Drug 
Drug  50/50  50/50 
No Drug  50/50  50/50 