aw when he stares at you, he is getting to know you!
babies love nothing more than to sit and stare at special people. the little one i am currently fostering is now 5 months, and at 3 weeks all he did was eat, sleep, poo, wee and scream lol. by 2 months he was a bit more active but not much, we just added puking to the list!
by 3 months your little man will be smiling, you might get a laugh and he will be stronger physically. tummy time is not important as long as you make sure he isn't always on his back laying flat. sit him up on your knee, hold him belly to belly.
talk to him, I whitter to my foster babies constantly, they love hearing your voice and it doesn't matter what you talk about, it could be the weather, or whats happening on Jeremy kyle or that you're just putting the washing on, whatever, the topic doesn't matter, its the tone you use. plus it will make him a good talker!
I have fostered lots of babies and raised my own kids too, the best advice I can give you is trust your gut. if it feels right, then your doing right and don't let anybody else tell you any different.
seems to me you are doing a fab job, now feet up, rest and heal. take this time to enjoy your baby because in 10 minutes, he will be 18 years old and borrowing your car lol xxxxxxx
p.s have you got a copy of birth to five? if not let me know and I will put a link on to the online version. it gives loads of advice plus it has a chart that gives a hint to when babies usually do things developmentally speaking.