Is it only the fork that dictates whether or not you use either a threaded or a threadless headset, or does the frame design also play a factor? Do they both just press into head tube, or are they mounted differently in the frame?

They do both just press in but I can't say I've seen any threaded forks over 1".

there were some 1 1/8" threaded headsets, and I think you can still get them and the quill stems required, but it's a quest. The only difference between threaded and threadless headsets are the threads in the top cup