It's not prescriptively grammatically correct, but it's the same thing as split infinitives or prepositions at the end of sentences. Native English speakers use the third person plural all the time anyway, and when the reason for your language's antiquated grammatical rules is wanting to be more like Latin, I think it's time for the pedants to catch up with reality.

As far as I'm aware, the use of 'they' for third-person singular of unspecified gender goes back centuries already. I don't see what the problem is.

The word you're after is "agender". So, pretty close.

The answer is - whatever they tell you to. Some would prefer "they". Others wouldn't care if you use "him" or "her". Yet others prefer some invented pronoun (there's a long list).

The only one that's likely to be inappropriate is "it". Don't use that.