You can see your movespeed (along with how fast your firing and reloading actions are) in the character screen, seen by pressing "@", or Shift + 2.
You can't see how fast enemies are, but most of them move and fire at 1.00 second delays. If you want to know how fast enemies are, you can check the DoomRL wiki, where their speed is listed in their respective articles. The important ones to know are that demons have a delay of about 0.8s, lost souls are like 0.35s, arachnotrons are about 0.7s and viles are about 0.6s. Mancubi are the only enemies that are slower than 1.0s, I think. You can't tell when their turn is about to come up in-game, you can only guess based on how fast they are.
As for the chance of successfully sidestepping an enemy projectile, that depends mainly on whether you're in running mode or not and the enemy you're trying to dodge. Mancubi, for example, will never hit a moving target but arachnotrons and former captains are much more likely to hit you once or twice even if you're in running mode and dodging about.
If you're actually trying to fight enemies out of cover without getting hit, you're not really thinking about it the right way. You should focus mainly on trying to kill enemies from around corners. Sometimes the cover and enemy placement wont allow this, and that's when you rely on your defensive traits by either dodging around if you have HR or just standing still and taking it if you have TaN/Iro.