so, he does karate three times a week, piano and guitar one night each a week and out for a drink with his friends once a week, and you're the one that is pushing him out?
Tbh I think men feeling pushed out after the arrival of a baby is not uncommon, often they can find it hard to not be the centre of your world. However, it takes two to make a marriage work, and by him being out 6 out of 7 nights and refusing to get a babysitter so you can spend time together, he's not exactly showing that he wants to do his bit to get back that closeness is he? But he's already laid the foundations of guilt at your feet so that if he does go off and have an affair with this woman (if he's not already having an affair with her) he can say that he felt pushed out by you and tht you knew how he was feeling etc etc.
This is what I'd do:
For now, I wouldn't say anything about the nights he goes out. But I would arrange a babysitter say for Friday night, but don't take dd to your parents, ask if they will come to your house and sit with dd. Then book a restaurant for about 7:00, and turn up to meet him from work all dressed up, take him to a bar for a drink and then on for dinner. (obviously he must know of none of this beforehand), and then spend the evening just the two of you, catching up and trying to get back what you think has been lost since the birth of your dd.
Once he sees that it is possible to have fun away from dd, he may be more open to the idea of going out more often. You don't have to leave her over night, I didn't leave my ds overnight until he was nearly two, some parents just don't feel comfortable with that idea, but there's a difference between leaving overnight and just leaving her for a few hours.
It is important for both partners to compromise to make a marriage work, but sometimes you have to use more suttle ways to make the man compromise, such as to make him see what he is missing out on and so getting him to think for himself.
Oh and I would mention the facebook thing. presumably you have your h and this woman on your friend lists which is how you saw these updates?