Does one who is studying torah need to wear something on his feet or can he do it barefoot?

The main thing is to learn as much Torah as possible, and if barefoot is more comfortable, then it's fine. Just when going to yeshiva, it's best to dress as accepted in that locale, so as not to attract attention, for we believe in the ideal of humility and modesty. Learning is not like prayer where we are, as if it were, standing before the King.