Temple of Ranma's Sailor Senshi Seifuku

Does the vice-principal of Furinkan have a canon name? I've run into quite a few different names in fanfic, but if there's already a name, why not use it?

No, there's no canon name for the Furinkan vice-principal. In fact, we canonically don't have any evidence that they even exist, it's just one of those background details that folks go with because it makes sense (like Ukyo's mother being dead, for example). About the only time we get any look in to the Furinkan hierarchy is in the anime; the Principal appoints Tatewaki as a substitute principal, leaving the school under his command, while taking some time off, and there's an old man in the basement who used to be the School Consigliere. We also see a group of people who may be the teaching staff or even the school officials in the anime episode where the Principal returns from Hawaii.

Does Principal Kuno even have a first name? I've read the name Kucho Kuno in several fics, but I can't remember him ever being named in the manga or anime.

No, the Principal is never given a first name. Even in the anime, where we get to see a report card of his from when he was a student at Furinkan (fun fact; he flunked every class he took bar PE, didn't get anything higher than a "D"... and one of the classes he flunked was English!), we aren't given one. I once saw him named "Groucho", but he's generally never named in fanon either.

As I understand it, the word normally used for his given name (Kucho doesn't look right, but I could be wrong) literally means "principal". It's his title, not his name.

Yes, he's called "Kunou-kouchou", kouchou being the Japanese word for "principal". Except for Ninomiya Hinako and Higuma Torajiro (anime only; the overly passionate gym teacher), none of the rest of the staff are given names. You can invent them as you like.

Really, in Japanese, names are less necessary because it's perfectly acceptable to call someone by their job title. The VP of the school is very likely to just be called "Kyoutou-sensei" - kyoutou being the Japanese title for vice-principal.