i do not agree with what ar81 said about this girl being too young. she is an adult, and we can't know how mature she is for her age. it is not fair to me to assume she can't know what real freedom is because she happens to be 20. of course i admit i am biased because i am often the youngest one in a group of people

but this is not as simple as ironically stomping one's foot about being an adult either.
to the OP i bet you are making this into a bigger thing than she is. she was interested in you for a lot of reasons, dating a music teacher would be cool and a lot of girls who date an older guy are a little bit proud that they were sophisticated enough to be seen on his level. that's how i was at 18 when i dated a 26 year old. having said all that, age really is just a number and it is only an issue if we decide to make it one.
it sounds to me like you are both committed to making healthier choices. if she wants to go to the bar, and you don't want her to go alone if she doesn't want to, you could go with her and order something nonalcoholic.
i definitely recommend that you both develop and maintain a few separate friendships. if this girl lost most of her friends when she and her ex ended the relationship, that, to me, is a red flag for potential codependency.
it is not healthy to draw all of your social and emotional needs from a single person, to lose your individual identity and become all about your partner. if she does have issues with codependency it is probably better to know now so you can watch for ways to encourage healthier ways of relating. relationships should be about wanting to be with someone, not needing to be needed. (truthfully i see a couple things in your own posts that make me wonder the same thing about you, but there is not enough there for me to be comfortable challenging you with that.)