pourquoi "i've got to go see my sister et non i've got to see my sister or i've got to go to see my sister,
I consider 'I've got to go to see my sister' the formal way to express it, predominantly used in written English: precise, formal and not really colloquial, synonym to 'I have to leave in order to meet my sister'. Some people wouldn't even say this in everyday life (to their peers). but it's absolutely correct and, as mentioned, perfectly appropriate in written English. Well, it's appropriate in spoken English as well, but, as we all have experienced, some vernacular expressions are common, even though they are not grammatically perfect. And conversely some grammatically correct expressions are considered more the 'sophisticated' way of speaking.
'I've got to go see my sister' seems to be a colloquial version, rather and not too rarely used in spoken English. Uttered during a party to explain your early leaving, nobody would take any exception to it, even not grammatically. But written down, an English teacher would pucker his forehead.(Well, let him)
'I've got to see my sister' is grammatically correct and expresses something different, that is, the insight into the necessity to visit your sister. Whereas the first two statements express the necessity to leave an event or a person in order to see your sister (mostly used as a pretext to leave!), this sentence merely means you've come to the general conclusion it's time you saw your sister.
By the way, 'I've got to' is, as far as I recall, a typically British phrase and, nevertheless, a little bit colloquial. But take this with a pinch of salt, I'm not exactly sure about it.
That's my view on this issue. I hope I'm not (too much) mistaken here and way off the mark.