Answers to your qs:
1. Yes, but it would not be advisable to spend money on a piece of land you do not, at the moment, own.
2. Yes and yes.
3. If it had been clearly pointed out to you that this area was incorporated into what you are buying and is evidently part of the curtilage of the house then I think any court would take a dim view of her now trying to claim some title to it.
4. Squattors title application and a deed of rectification- if I were giving you a quote I would say 2,500 to 3000 inclusive. Check with your solicitor.
5. Yes, go to the actual registered owner ( ie the person previous to the person who sold to you) and ask THEM to do a deed of rectification. If they agree the whole thing is shortcircuited, much cheaper and easier and a guaranteed result.