Due to a contractual situation Marvel are not allowed to use "Mutants" in their movies, and 20th Century Fox are not allowed to use The Avengers characters in theirs.
How this is exactly detailed in the licensing agreements I'm not sure is entirely clear, but what is apparent is that there are some 'grey area' characters, including Quicksilver and Scarlet Witch.
In the MCU, the Maximoff twins are not mutants, and they are not children of Magneto. Magneto does not exist in the MCU.
In the X-Men movie franchise, Peter Maximoff is a mutant, and is clearly aware that he is the son of Magneto, but chooses not the share that information. There is also no mention at all of a twin sister.
Nina, the much younger daughter of Magneto, is clearly not Peter's twin, isn't called Wanda nor does she appear to share any powers with Scarlet Witch. I don't think we're meant to assume that she is that character.
Why did they choose to kill off Quicksilver in Avengers - Age of Ultron?
There's no indication that this was due to a rights problem as there has been no dispute over Age of Ulton or the two X-Men movies with Quicksilver. They are clearly in a position where both organizations can use the Quicksilver character.
I think the character was killed off for plot purposes. He is a 'second tier' character in importance, having only appeared in post credits scene prior to the movie. The Maximoff's coming over to 'the good guys' side and his (eventually friendly) rivalry with Hawkeye is an key character arc in Age of Ultron.