I've been meaning to do a video on the difference between pages, IA mode, presets, songs, etc. Until I can get some time to do that, I'll do my best to explain.
There are 16 pages, and each page can have up to 22 buttons, depending on your model. Any of the buttons that are not IA buttons (or the IA Mode button) can have a second function "underneath" its main function. This second function is always an IA. Pressing the IA mode button and turning it on reveals the IA functions underneath the main functions.
IA mode is separate from button page selection. Pressing the "Page / IA Store button" selects the currently visible page and pressing the IA mode button reveals the IA functions on whichever page is currently visible.
Some answers to your specific questions:
1). I'll check into this issue, but it should definitely wrap around to the first page after reaching the last page. There is a global setting that determines how many pages are accessible, and it's set to 2 pages by default. Maybe this is what you're seeing?
2). In the editor, there's a setting in each preset called "Show Page". Set this number to the page you'd like to have automatically displayed for each preset, and the GT will automatically switch pages for you when you switch presets.
3). I don't have a GT and/or Axe-Fx in front of me now to check, but I will check this when I can.
4). To make the Page/ Store button stay on in IA mode, remove any button actions from that button. If a button does not have any button actions, it won't change when IA mode is enabled.
Give all that a try and let us know how it goes!