In this category the rules don't say you can't do it on second quest, and if you do it on second quest it would be faster because the koopas move faster. Am I wrong?

It would probably be used already if it were allowed so I pretty much doubt it

It seems that Second Quest can indeed be used, see https://www.speedrun.com/smbce/thread/1tnk1/1#mgxds
"Level select / second quest is allowed in categories that do not beat the game ..."
But:
"All categories begin in first quest, except for "Second Quest" and "Second Quest Warpless ..."
So you would have to get to Second Quest first and the time for that (at least 2:40 min if you do Minus World Ending) would be added to your run time.

K