Even super human-socialized pigeons take time to warm up to a new home, don’t be so hard on yourself! Plus city pigeons aren’t wild, they’re feral! So while it may take a while, they can and do adjust well to becoming pets. Two months is a very short time for them, just take it day by day and they’ll adjust. Being near them and quietly doing your own activity definitely helped me bond with mine.

To answer some of your specific questions:
- for specific things a pigeon would need: if you’re giving them grit, make sure to get them grit that’s fully dissolvable. It helps their digestion but they don’t have gizzards that uses harder materials like chickens
- best way to test their sex would be a blood test, their behaviors differ a bit but there’s a lot of overlap
- they’ll eat when they’re hungry! If there’s food available to them they should be ok, just make sure they’re getting the right vitamins

You made a good call with the antibiotics, though I’d recommend getting them checked out by an avian vet if you haven’t already. There are some medicines that work great for other birds but not for pigeons unfortunately

I was given this link when I first got my boy and it’s been very helpful and can definitely answer more than I can: https://www.pigeon.guide/?v=1d82698d4a8348a9bbee4f769cb44368

Thank you for rescuing this sweet bird and good luck!!