As large as possible. At the moment, I just have a little seven inch tablet, and it's nearly impossible for me to use for anything, because of the various impediments like "screen hogging" applications (like the PDF reader you mentioned) and I find the touch screen to be unwieldy.
I had an iPad I guess 10" or whatever it was, but it died on me. The size was OK for personal, up-close use, like reading "books" and web-surfing, but it's pretty small for using at a distance, like one would when playing music. Just imagine if you had a physical book in those dimensions, and add in all the software cruft: would be OK for just glancing at to refresh your memory, like a jazz chart, but pretty small for detailed work. I don't think the backlit screen is necessarily better or worse for the eyes, just a matter of getting used to it. For extended periods of use? Yeah, I get eye-strain from computer screens, but I wouldn't know how to mitigate that other than using some of the "color temperature" apps to add/reduce some of the "blue" wavelengths out.
Page turning? At least for the iPad and likely any Android tablet there are some foot pedals one can use. I don't know any more than that other than that I've seen people use them.
TBH, if I were going to drop a bunch of cash on a new tablet, I'd be inclined to look at a lower-end "convertible" notebook computer. When I was last looking for a new notebook computer, the prices were almost shockingly reasonable, including even the Microsoft Surface models.
So, you can have the tablet mode for live performance, but at home you can still use the keyboard/trackpad interface.
I just bought a cheap music stand (about USD 40) which can be locked into place as a flat surface on which a notebook computer could be placed. I just use it as a regular music stand, but that's an option if you don't want to balance your computer on the edge of your keyboard/piano.
That's all I know, as I still prefer to print stuff out if I'm going to play it, or use an old-fashioned bound printed matter book.