It's always better to answer yes if you're unsure than to answer no and be found incorrect. You can always explain the circumstance if it comes up later.
That being said...
For 1, paying a fee to avoid miliary service is not considered having served in the military - you were exempt from it. Weapons training, you said you don't have any, so this would be no regardless. This is only different if for some reason paying the fee gave you a military record, but it shouldn't.
For 2, the answer is definitely "no" for speeding tickets. Traffic violations are civil, not criminal unless there's a DUI, reckless driving, fleeing an accident, etc.. serious crimes.