"Can the PSU be making my computer slow? " No. As long as you have enough power to post and boot, you have enough power - unless you are doing 3D stuff. Power demands of a modern video at least double when shifting from a 2D desktop to a 3D gaming load. So a flaky or inadequate PSU might work until you try to play a game. Usual symptom is seemingly random reset/reboots during 3D games.
"Dying PSU could also be why my DVD burner screws up often? " Now, that's possible.
While I agree with jsc, it is vaguely possible for a marginal PSU to cause this IF the system keeps encountering errors in certain components that are temporarily "fixed" by automatic re-trys. The best proof would be to replace with a different PSU of at least 300W capacity, even if only temporarily, to see if that helps.
There is also the possibility that some other part of your system is faulty and causing this, and may even have caused trouble for the original PSU.