Do you allow takebacks in rated games?

They won't learn much by doing takebacks. Let them just resign, be a man and play another game. I played this older man, he said he took medication for some mental disorder. He asked for a takeback, back then I let him. Several moves later he asked again for a takeback, I let him. Several moves later he blundered and lost a R and purposely tipped his K over, signifying he resigned. Then he looked at the board, saw a possible win at that point, stood his King back up and asked for takeback. I let him and he went on to finally beat me. The game took an extra 45 mins. Then it was me who was a little upset and I'm sure it showed. Seeing that, he left in a huff. Maybe to go take his medication. I don't allow takebacks any more.

Take-back is for children; no self-respecting player should even think of playing this, not even in an unrated game.

I accept takebacks only in casual and only for obvious mouse slips

No, absolutely not. Part of learning to get better at chess is to not move until you are absolutely sure of your move.

Feeling bad over a non granted takeback to a guy who calls you "a" "women" (...) i.e. makes a stupid sexist insult to you, that's really weird. Just crush him, afterwards report for inappropriate behaviour.

I usually don't accept takebacks in both casual and rated games. If the other opponent blundered, it's too late. I may accept them if the opponent did an obvious mouse slip.

I only ask for takebacks myself when I mouse slip, which is usually playing Kf1 (or Kf8) when I want to castle. And I always accept them in that case. This has nothing to do with chess so this is basic sportsmanship in my opinion.

I usually accept them also if I think my opponent made the best move. If you want to change it for a worse move, be my guest.

I sometimes accept them when my opponent blunders. I know I shouldn't but blunders ruin the game for me and it doesn't satisfy me to win because I just took a piece that was hanging or something like that. I never accept them when my opponent failed to punish a blunder of mine and then ask a takeback.

I never accept them in time trouble either. On the other hand I may accept them in correspondence games because to me those are about the quality of the game so if my opponent blunders a piece I feel I wasted my time.

I consider it basic sportsmanship to use the move confirmation option and make sure the move is what you intended before submitting it rather than just hope you get it right and expecting your opponent to clean up your mess if you make one.

Confirming every move takes too long though, especially in shorter time controls. I guess you could have it on for longer time controls though. Just this morning, I played this game, lichess.org/dqvLWIgP/black#16 where I was planning on moving 8... Bc5, centralizing my bishop and preparing to castle, but misclicked d6 and hung my knight.

Yes, if you are trying to move fast, then it is even worse to try to have your opponent try to clean up your mistakes. That should be obvious.