1) What can I expect to happen in both locations? How long do they last? Service starts at 2
Usually a crematorium service is relatively short, IME, and they tend to be strictly scheduled as there will be one before 'yours' and one pretty immediately afterwards. I'd certainly not expect it to run on for more than 40 minutes.
Will the pub wake run into the late night? Will people drink a lot and try to have fun, or just have a quiet couple of drinks and head home? Will there be food? I have food intolerances but don't want to bother the hosts about these and they didn't ask. Not sure how to handle food - should I leave for a bite at some point and come back?
The length of the wake is impossible to predict! But usually they are very positive and somehow celebratory events. Some people might leave after a short period of time, some may stay and party. You just do whatever you feel most comfortable with (especially if you don't know anyone else there). There's no obligation to stay for a set length of time.
Re the food, yes I certainly wouldn't bother the hosts. If it's in a pub, most likely they will have laid on things like sandwiches. If I were you, I'd aim to have a big breakfast so I wouldn't be too hungry at the wake - and then just play it by ear when you see what's available.
2) Invite says to be yourself when choosing the clothes and to channel your inner rock chick (she was, and I am to some extent too). To just dress like you used to when you went to see her when she was still alive. Is it still best to dress in dark conservative clothes to be on the safe side?
What they're basically saying here is that you don't need to turn up wearing black from head to toe. Wear what you feel is comfortable, and that reflects the relationship you had and times spent with her.
3) What should I definitely bring? Should I give anything to her family?
You don't need to bring anything or gift them anything. There will most likely be a collection plate at the crematorium, for a charity of her/their choice, so you could give cash to that. I'd generally aim for £10-20, depending on how close you were. (Obviously assuming you can afford that!)
4) I won't know anyone at the funeral and wake. Should I try to speak to people to introduce myself and mingle, and share memories of the decesead and so on - what is the etiquette there?
No particular etiquette, but yes try and mingle if you feel comfortable.
And I'm sorry for your loss 