] ...Mrs. Hughes asking Catherine to "let in" Miss Tilney next to her at the dance. Did a new party joining into the dance have to be introduced in?

No, but normal ediquette was to join the set at the "bottom". That is, farthest away from the music, and the last to begin dancing. (You had to wait for the "lead couple" to move down from the "top"; it also meant you were the last to finish the dance.) Otherwise, you're "butting into line", to use a modern analogy.

] Later Isabella seems to use the "set" in terms of the how the dancers are grouped together, rather than in terms of a two-dance set, it seems.

Yes. A "set" is all the dancers in the two lines (men on one side, women on the other, facing each other). You might have another set running parallel to that one if the room is wide enough and/or the room is to short to accomodate everyone in one set. There is also the "minor set", the two or three couples needed to perform all the movements of a particular dance. (Dances can be catagorised as "duple-minor" or "triple-minor".)

I remain,
Jason E.