To get Time Hell you actually have to get it down to 2:44, and *not* merely 2:45 as the medal description falsely states!
Actually for all the time-based medals you must make one second less than stated in order to obtain the medal.
As to how to do that, first be sure you absolutely have the fastest method of completing the stage, for all stages. Are you sure? It may take a few tries and you'll find a slightly faster method. Remember every tiny miniscule amount of time counts, so shaving the tiniest amount of time off a stage will add up, especially combined with all the stages. Be extremely risky with enemies because if you shave a few pixels off that adds up to time savings.
Also just replaying and doing the exact same thing sometimes is a tad faster or slower on some runs, so you'll need to keep replaying through a bunch.
Medals don't pop up in-game but will be on your account if you earned them.