First and foremost, your shop should have some sort of policies in place in order to deal with these scenarios. If it doesn't, I would get crackin on that ASAP.
And now onto how I would have dealt with these problems:
1. I would charge them for an americano. Thats two sets of dishes you have to wash. If you have milk out for customers to doctor up their drinks, you should probably expect them to.
2. I would refuse. Simply put, this is in the same realm of people asking if they can pour their own drinks (there's a thread on here somewhere about this very situation), which is rude, degrades your position, and shouldn't be tolerated. We don't sell hot water, we sell tea. This lady should, theoretically, have hot water at home.
3. The problem here is "tea bag" and "for here". If you have a loose leaf option, I'm all for people sharing tea.
I would say that you shouldn't be complaining about these scenarios when you (your company, not you personally) have done nothing to rectify the problem.
Fraser, try this: Take a new cup and jacket to her table, and ask her if she would like a new one, as the one she is using is last year's model. I am curious what her response would be... :)