Congratulations! You did it! You had the baby! Your journey was successful, and it sounds like you have a healthy baby - such a blessing.
Now for some tough-love, my darling. Any relative that will take you: hormonal, post surgery and with an infant should be appreciated. It is not easy to have an infant in the house. It was one of the hardest things I ever had to do, and most of us have to do it sleep-deprived, which adds a whole new level of unreality to the situation.
I don't think any of my relative would do that for me. Appreciate anyone who does it for you.
Relatives and friends who are parents are going to give advice if you are a first time Mom. It happens to all of us who become parents. Nod and thank them, even if you don't like it. Concentrate on how nice it was to get a ride to the peds and not on the comments you don't like. Your relative took time out of her busy (and yes we're all busy) life to go with you! Soon enough you will be on your own - believe me - you'll think whistfully about how nice it was to have help.
A Friend who call your child "our baby" mean nothing negative buy it. My guess is that it's a way of sharing her excitement over your new baby with you. I wouldn't read into it. Wasn't it nice that she came to the hospital to see you and your baby?
Count your blessings, overlook the frustrations. Your daughter will not be an infant forever: she will sleep through the night, then talk, then be out of diapers, then off to school...enjoy your time now.