When expressing "Going to" in English we use "Mynd i" in Welsh so "Dw i'n mynd i aros" (I'm going to stay", "Dw i'n mynd i Ffrainc" (I'm going to France) and "Dw i'n mynd i fynd i Ffrainc" (I am going to go to France).
I tend to think of them as different 'to's. One is an infinitive marker, the other is a preposition used with "to go" to form the future aspect. And yes you are correct except "i" causes a soft mutation so "Dw i'n mynd i ddysgu Cymraeg".
Yes, in some contexts in some dialects. It is not very widely used, though, and not generally taught on introductory courses. There was an example which had popped up in one of our examples, but it has now been removed in order to avoid confusion.