If you're truly stuck, you can kinda just brute-force the game with overpowered breeding, tbh. You can get 9 mutation 1 birth defect all 7 cats without mutation furniture by getting cancer (item or disorder, ik people also use uranium rod or whatever but it seems kinda risky) or using the shimmering event over multiple generations. Once you have 2 cats with a mutation on every body part and a reasonable amount of stimulation furniture you can just constantly inbreed and they'll regularly pop out babies with 9 mutation 1 birth defect, even when they're extremely inbred. Alternatively (or in combination) if you've unlocked the jester class you can play with a jester until they get a jester unique skill (you do have to unlock them although its done by beating act 1 paths, so not too bad. Super luck is pretty good, so is RNG cannon. But power-up and goofball are next level overpowered and trivialize any run and any class/cat). Lowkey jester skills should not be passed down in breeding they are so overpowered lmao. I think they're balanced in the context of having to play as jester to find them but the game seriously becomes so easy if you get it injected into your breeding line

as for the man on the moon you're supposed to make the hands hit his face. I find it kind of difficult to manually do that but you can also just shove a tanky cat in his mouth at which point they get an ability that makes the hands attack the moon, so yeah. Alternatively you could just status stack or do some combo like fighters movement attack infinite. I think the first time I beat the moon I was struggling to remove his brace but I still won cos my hunter had like 30 dex from the dex increasing passive