I doubt you'll find a soft underlayment that's that thick, 1/2". You will probably need to lay down 1/2" plywood and then add your flooring on top of that.
There are many reducer transition pieces if you should decide to keep the new floor lower than the existing one. I know you stated you don't want that but it might end up being a viable ...
Self levelling low-spots only, and optionally adding plywood to smoothen the rest.
"self levelling" cement is not liquid as you think and you can apply it in patches without worrying that it will run like water. You have some troweling time to work low areas as large as say 5x5 ft.
No, there isn't. You'll want to follow a common strategy of either running a full tile or a joint down the center of the room so that the rows at the edges look appropriate. Snap one or two chalk lines on the floor as references, then snap one or two at 90 degrees to get your joint lines straight.
With a wet saw these types of cuts are not difficult nor time ...
If your plywood floor is in good condition, I would drive screws to securely attach it to your floor beams. When I did my floor, I saw they were attached with nails which was not ideal even though I had no squeeky issue.