They are being very weird to care so much, but you also sound a bit precious about some of it. Most people look fairly close to their age, it's pretty unlikely you pass for 35, they can probably guess you're between late forties to early fifties so does it matter if they know/don't know that you're 50 exactly? Same with your birth date, it's odd of them to want to know but also a bit odd that you're so keen on them not knowing -how does it actually have any negative impact on you if your colleagues know you're born on July 2nd or whatever?
If they have been making enquiries about things like your health (e.g. a reason for why you've been off sick) or sexuality, then that's not on, although tbh I'm not sure what HR could really do if it was out of work, but tbh good friends wouldn't pass private information on to third parties anyway, surely?
Not getting involved in gossip and keeping work/private life separate is completely fine, but depending on how you word your refusals I can understand why it might come off as a bit odd. e.g. if they are chatting about 'Ah it's rubbish having a birthday so close to Christmas, I never get 2 sets of presents! When's your birthday Tim?'
'June, still bloody rains every year though! How about you, Mary?'
'I'm not telling you.'
Is a bit odd.
Similarly if someone asks "Are you married Mary?' There's a range of options between
a) "Well I was married to Bob Jones, we met at school and were together for twenty years but then I had an affair with my next door neighbour, realised I was a lesbian, got a divorce and now I'm on Tinder making up for lost time."
b) "No, I'm divorced, but enjoy the peace to be honest haha! Do you have that spreadsheet we need?"
and
c) "That's none of your business."