The Doctor would know better than to fight Bugs Bunny.
Princess Celestia as she is now couldn't personally beat any of them, but she'd send Twilight Sparkle to do it, and Twilight could "beat" any of them except Bill Cipher by using the power of friendship (which is to say, she could convince any of them but Bill to be friends with her and her friends, or she could deploy Fluttershy to do it.) Twilight could not personally win Bill Cipher over with the power of friendship or get one of her team to do so, but she could get Fluttershy to get Discord. Discord might be able to beat Bill, or might get his ass kicked, it kind of depends who's writing it. Against Twilight (sent by Princess Celestia), Q, The Doctor, Steven Universe, and Bugs Bunny all decide to be friends and don't want to fight anymore, which is winning in a sense.
Steven Universe also has The Power Of Friendship, so using that, he'd make friends with Q, The Doctor, Princess Celestia and her minions, and Bugs Bunny. The only entity malevolent enough for this not to work on is Bill Cipher.
Bill probably beats Steven, cannot beat the Doctor (the Doctor is specifically designed for fights against Bill-class entities), can beat Princess Celestia maybe even if she pulls in her minions and they deploy Discord, cannot beat Q, and cannot beat Bugs.
Q could beat any of these except Bugs. However it is unlikely that Q would take a fight with Bugs seriously, so he probably "loses" because it's funnier that way (also because the way all stories are written, Bugs must win.) The Doctor is also designed for beating Q-class entities as well as Bill-class, but a fight with Q generally redefines "winning" so that maybe Q has been driven off, and has stopped doing whatever he was doing, but then it turns out he was doing it to make you learn something you didn't want to face, and the Doctor is as vulnerable to that as Picard is. I would say that by that metric, Q "beats" the Doctor because Q wouldn't get into a fight with the Doctor unless he had something to prove, and he'd end up proving it.
So actually the only real fight here is Bill Cipher vs anyone who is neither Bugs Bunny nor the Doctor, as all the rest of the contests either end in friendship and mutual agreements, or Bugs Bunny wins. We know the Doctor can beat Bill, and Q can beat Bill, but Bill vs Celestia (and team) or Bill vs Steven (and team) are real fights.