How much did you pay for it? Where are you located? Do you know if it's a good battery they put or a not good battery?
it was like 30 euros or something, and it wasn't original as far as I know, but new, it doesn't drain in half an hour, so it's as 'good' as it gets.
I don't think Apple would have parts for 'obsolete' devices, you should check third party repair services.
I have to check which IOS I'm on with iPhone SE 1st Generation. I know it isn't the latest version though. So should I update it or not? Or just check what version it is first?
as far as I know, updates stopped at 15.8.5, and there might have been security patches since, you should get what's available.