How Are My Stair Openings Framed (60s side-split)
Wow, you even opened up some of the wall to help determine which way the floor joists are running. (Often this is done with a “stud finder”.) 

I see nothing unreasonable about your joist layout: 1) joists follow what you discovered when you opened up the wall, 2) joists span the short distance across a rooms, 3) joists have “normal” span lengths, (nothing excessive)

However, I’d guess the joists were doubled where joists change direction. 

To answer your question about how stairs are framed, I’d go to Architectural Graphic Standards. (You can Google it.) Typically, 2x12’s are notched for the proper “rise” and “run” called stringers. I’d guess there are 3 stringers: 1 on each side of each stairway and 1 in the center.