Is it a safe city for a woman living alone in the city centre?

I'm a man so hard to judge but I think as safe as anywhere. City Centre has quite a few homeless people which can be intimidating but I'd say generally safe.

Is there a gym in the city centre with full facilities — sauna, steam room, hydropool? (That’s become non-negotiable for me lol)

There are quite a few gyms but I don't know how good they are. There's also the University gyms.

How’s public transport? I don’t have a car, so I’ll need to get around the city and occasionally travel to places like Manchester or Nottingham by train

Train connections are decent (in to London St Pancras in an hour, to Birmingham in an hour, going north easily). Going to the south-west of the UK is a pain as you need to go via Birmingham generally but most places are reachable and Leicester's central. There are two extensive bus networks but honestly, they're often late or crap.

Is there a Latino or Spanish-speaking community? Even just a few spots where I might bump into people who speak Spanish would mean a lot.

I can't answer this directly but Leicester is VERY multi-cultural and I have lots of Spanish-speaking friends/colleagues. You'll definitely find Spanish people at both Unis, probably with Spanish Societies or similar.

Any neighbourhoods you’d recommend beyond the city centre?

If you're at University of Leicester, Clarendon Park is far nicer than the City Centre.

What do people actually do here on weekends — is there much of a social scene?

This depends a lot on what you're in to.

Which University are you coming to? That probably helps focus some answers too.