Well, I feel like they've gotta be able to somehow control their perception of time, to avoid this problem.
It's like that old Einstein saying: "Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute." The speedsters are just able to mentally manipulate this relativity.
No. While not moving at superspeed a speedsters mass is the same as a normal person and perceives time at a normal rate. Once a speedster is moving at superspeed, the closer he she gets to the speed of light, the greater their mass becomes and thus time does slow down for them.
I remember a story from Don Rosa where Donald Duck got super powers, including super speed. He wanted to run around the world in two seconds to show off, but since his mind was sped up too he realized it would still feel like years for him, as if he had walked at normal speed, so he changed his mind.