I got the Ultima 7 collection here on GOG. Do I install and finish the Black gate first, and then when I am done install and play part 2 Serpent Isle?

Or do you install both and Black Gate takes parts from Serpent isle?

Also, if I use Exult, will it apply to both parts one and two? Thanks - sorry for the questions

The two games are entirely separate. Not sure why they are called "Part 1" and "Part 2" instead of just continuing the numbers, except maybe they chose that naming convention because the two games utilise the same engine.

Story-wise, you're better off playing Black Gate first and then Serpent Isle after, but actually there was another game (Ultima Underworld 2) with some additional story that fits chronologically in between U7 part 1 and U7 part 2. However, it's very minimal and nothing is really lost by skipping Underworld 2. Yes it does - you can use Exult to play either game.