Run speed: no, they don't. For the mostpart they run about the same speed, which is about as fast as you are, but there are some creatures I can run from without being clawed in the back and some that just mow me down if I try to run.
Leashing: if you are fighting or being chased by a creature, and you get too far from its spawn point, it will suddenly stop chasing, dash really fast back to its spawn point, evading/resisting all damage as it does this, and regenerate its health quickly as it resets itself. The implication of this is that you can't train people as easily and you can't kite creatures all over the place. Creatures will often follow *longer* if you continue to build hate, but all creatures will leash eventually. (or are supposed to, anyhow.)
Soloing: is possible by any class. Warriors and rogues tank things to death and munch on some food afterwards to recover their health. Paladins and priests tank while dealing damage, and heal through the fight. Mages ideally never get touched. Warlocks have their pet tank. Druids and shamans do any or all of the above. Some are slightly better than others, but they are all competent. You won't be able to solo every quest, especially the ones designated to not be able to, but soloing is very plausible. Some people who never learn to group think of the game as Morrowind with other people running around.
Speed: nothing permanent, really. All races can get a mount at level 40 except for Tauren who get the racial trait plainsrunning instead, as was mentioned by the people before me, but these aren't combat-effective. A few minor bursts of speed for escape purposes also exist, but nothing like jboots or innate run speed.
Only the exceptions can be exceptional.