I guess you have to choose your battles. You can't fight every single infraction and need to let kids work some of it out for themselves.
And I agree with you that there's a difference between seeing behavior as common at a certain age, and allowing it to continue. They do it, not because they are evil, but because they don't know any better or because they get a pay-off for it, or both.
Some things you can let them grow out of. But other things require your intervention, and your daughter's need to get her own way right away all the time is a problem. So perhaps you choose 2 things to work on within the family - privacy/door opening, and hitting her siblings or others. Those have to have immediate and unpleasant consequences for her - and apologizing isn't one of them. First of all, kids this age aren't really sorry a lot of the time, so an apology is perceived by them as a "do-over" - something that makes their prior action acceptable or at least nullified. So if you make her apologize to the sibling she hits, or to you for opening the door, it a) gives her all the attention and b) doesn't stop the behavior the next time. So immediate and negative consequences, with zero attention, might be more effective especially for someone who wants to be at the center of things. So, even though it's more effort for you, immediately focusing on the child who was hit and banishing her to her room (with as little discussion as possible) might be more effective especially if you do it every time.
I think you did the right thing making her search for the other kid's hat. I think requiring her to apologize probably had some value since it was a public situation and there may have been some expectation on the part of the other family, but as you noticed, it's hard for her and she doesn't really mean it. And she did get a pay-off for it - she wanted more attention, and she got it. So perhaps one opportunity to apologize followed by some isolation/loss of attention might have worked. And if she refused, some increase in the amount of isolation would be effective. Perhaps sitting by herself off to the side, or even in the car, would be effective. She's simply not welcome in the group until she can act more kindly and politely. Now, if she's the type to jump out of a "time out" situation and thus require even more of your attention just to keep her sitting there, then it's not going to work. A lot of people would have paid more attention to the 11 year old at that point, asking more about the hat and what his favorite team is, seeing if he wants to throw a frisbee around, that sort of thing. Sometimes that works great.
But you are wise to notice that she doesn't see the consequences of not being able to go back to Family A's house if she is rude, that she can just go to Family B, C, D, and E the next time. You have to disavow her of that notion immediately. Maybe she has to be taken home immediately, maybe she has to be given a trash bag and sent off on the other side of the fence to pick up any litter, etc. And the answer to requests for stickers or dresses has to be "No, those are for helpful and appreciative family members." The older kids get, the easier it is to defer consequences, but that's a progression, and telling a kid they can't do something next weekend because of an infraction today often doesn't work. I do think some sort of public response is important in a public setting, so a sharply worded "That is simply not acceptable, Susie!" is a start, followed by excluding her from something (making her go last in the buffet line or taking away the chance to play croquet, whatever is going on) lets her know that she cannot be trusted around other people where manners are required. She can always have the opportunity to prove to you that she knows how to behave, and that can be achieved by some private time while she thinks about how she's going to act differently.
Sometimes with oldest kids, there's a certain amount of bossiness and a need to assert their position in the family - and you can't intervene all the time. She may be reacting to the new baby too. Yes, older kids have more responsibilities, but they also have more privileges. Being a role model and not a source of nastiness to her younger sibs should have certain rewards. It's okay to let her do more things than her younger sibs, but if she teases them about it (who got a play date, who got more chocolate), then she's showing that she's not mature enough to have treats/privileges, and I'd tell her so. She gets more when she is kind and when she opts not to hurt someone's feelings.