People here often grumble when a questioner hasn't done enough work on their own before asking us to help. But in your case I think you've done too much!!
Bronte is doing something unusual here. Imagine you've been looking after a neighbour's child all morning and he has been driving you mad. When telling a friend about it later you might imitate the child's whine and say, "Why haven't you got any Coke? Why can't I go to Dweezil's house? When will Mum (Mom) be home?" Or you could put on exactly the same voice and say, "He wanted some Coke. He wanted to go to Dweezil's. When would his mum be home?" Have you heard that being done? It is a bit illogical: the child didn't actually say the words "He wanted" or "his mom". Nonetheless, I've certainly heard myself doing it.
So. The aunt didn't actually say "She regretted..." Bronte uses the speech marks (inverted commas) in such an unorthodox way because she wants the reader to imagine her aunt's voice.
What her aunt actually says is something like this: (You're probably way ahead of me and don't need the rest, but I'll do it anyway!)
"I regret being under the necessity of keeping you at a distance; but until I hear from Bessie, and can discover by my own observation, that your are endeavouring in good earnest to acquire a more sociable and childlike disposition, a more attractive and sprightly manner-- something lighter, franker, more natural, as it were--I really must exclude you from privileges intended only for contented, happy, little children."
A couple of things you queried:
to be under the necessity of: needing to or having to
endeavouring in good earnest: trying seriously
from privileges intended only for contented, happy, little children:
not "from things that only kids enjoy" but from things only children who act like children - proper children, in her view - deserve.
The aunt sounds pretty loathsome. As you say, she and Jane clearly dislike each other. I should read it.
Best Answer
The phrase "get to do something" implies to have the opportunity to do something. So the meaning of the first sentence is "I love to have the opportunity to see her first thing in the morning".
The phrases "get to do something and get to doing something" aren't interchangeable to convey the same meaning. The latter phrase means "to start doing something such as I got to talking to her. I got to thinking that it was his fault.
Both the sentences are grammatically correct, but they convey different senses.