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These are all reasonable questions. I'll answer as best as I can, but please understand that I'm not any kind of formal authority.
> If you cannot use requested pronouns, does your religion also ban you from using nicknames? Is it okay for you to be called User23?
I don't know of any Catholic doctrine that says people can't use pseudonyms, nicknames, or even change their name altogether.
> Is it okay to say someone's cat is cute even if you don't really care for cats? What do you do if your partner asks if they look good or bad in something?
Is it a lie, which is to say a falsehood told with the intent to deceive? Then yes it's wrong. Wouldn't you want to be told the truth if you in fact looked bad? Wouldn't you want to know that when you're told that you look good that you really do?
Personally, supposing I didn't think the cat was cute, I wouldn't say I thought it was. I would most likely treat it as a good opportunity to say nothing on the subject. Some theologians put forth a doctrine of "mental reservation"[1] which somehow makes lying OK, but I have to admit I'm not capable of the necessary mental gymnastics in any but the most clear cut cases.
