After looking over the rules, I was wondering, are there any hard rules / dos / don'ts as to what pet names aren't allowed, and which ones are more acceptable? If so, what is, and isn't, allowed? i.e. are potentially controversial names allowed, or no?

Other than the obvious swear words - we can't say much. If you're considering a borderline/controversial name I strongly recommend you make a ticket and ask staff directly.
We can give all the advice we want, but ultimately its staff who interpret the rules. I wouldn't want you to get into trouble because a random user figured a certain name was okay or not.
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I talk from experience: ANY name is allowed, I had a pretty offensive one but I filled a ticket right after creating to let staff know and left a note in the profile about what the pet would be about, so people would know it was for pure offense. You don't have to worry too much about sensitive people seeing the name because they usually use bad word filters on subeta since swearing is allowed everywhere.
Any name is not, in fact, allowed. We actually have a rule about inappropriate pet and minion names and encourage anyone wondering about a specific name to submit a ticket before creating a pet that might potentially land them into trouble.
lmao that's definitely not true... remember when the pound was full of pets with names like kumos_5789473 and feli_798310? that's because those were pets with 'obscene' names (extreme swears, slurs, etc) that got renamed by staff. like, hell and crap are okay, but notice how there's no pets named fck or sht or c*nt?