Where does Elizabeth get her powers from?
As the titles says, Elizabeth has certain powers through the game which are a part of her
while the Lutece siblings need science and machines to achieve the same results
Why does she has those powers, and, is she the only one who has them, or are there others?
Potential spoilers below.
Rosalind says that it's probably because part of Elizabeth exists in another universe (her cut pinky), and thus can interfere with the tears/make new ones.
The actual quote: "What makes the girl different? I suspect it has less to do with what she is and more to do with what she's not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge."
As for the Luteces, when Fink sabotaged their tear machine to kill them, it somehow caused the siblings to be able to travel through tears freely, which is what they do throughout the game.