which is in contrast to his son, Henry (although I had to pause on that thought when they went through Henry's designated room with his "guns, and greatcoats"...)

two things popped out to me regarding the General's guided tour of the Abbey with Catherine:

One was about the library:

they proceeded into the library, an apartment, in its way, of equal magnificence, exhibiting a collection of books, on which an humble man might have looked with pride. Catherine heard, admired, and wondered with more genuine feeling than before — gathered all that she could from this storehouse of knowledge, by running over the titles of half a shelf, and was ready to proceed.

The General does not have a particularly vast library. Ok, it was compared to the drawing-room that only was used when important people around, but I get the impression that he may have time for pamphlets, but not for books. I think the narrator was being facetious regarding any magnificence whatsoever. General Tilney is not a humble man, so he can't look at his library with pride, nay? Catherine (again, because they weren't OLD books perhaps) shows very little interest by looking over only half a shelf.

The other thing was about the billiard room.
[Catherine] observing several doors that were neither opened nor explained to her — by finding herself successively in a billiard–room, and in the general’s private apartment, without comprehending their connection [...]