“Enough” is an approximate translation of dóthain; a more precise translation would be “sufficiency”. Thus, the original sentence would translate as “Do you have your sufficiency?” Without the do, it sounds something like “Have you had fill?” would in English (i.e. if the “your” were removed from “Have you had your fill?”).
The a here is the possessive adjective “her”. A literal translation of Níl a dóthain aici would be “She doesn’t have her sufficiency”, and its meaning would be “She doesn’t have enough”. Like a Níl sentence, a Tá sentence would also need a possessive adjective, e.g. Tá mo dhóthain agam.
dóthain is a noun and it is used with a possessive pronoun, so it's do dhótháin or mo dhotáin, etc. The closest equivalent in English might be "your fill" (though you would probably change the tense a little as well.
leor is an adjective, and, while it is also used in phrases that indicate "enough", it is grammatically quite different, and there can be a slight difference in nuance as well. (While the FGB describes leor as an adjective, it is used in the copular construction Is leor sin).
They are not interchangeable, though you can often create two statements that mean much the same thing, one statement using leor and the other using dóthain.
Well, no. Imagine, if you will, your hostess holding a tray of canapés (not French couches but Irish mushroom thingummies), into which you've dived with enthusiasm, gobbling up three-quarters of them and sloshing her excellent white wine down your gullet after them. She fixes you with a cool gaze and asks, well, "An bhfuil do dhóthain agat?" in other words, "Have you had enough?"