"All you have to do is press the button." The extra "to" must be omitted. "All you have to do is pressing the button." is completely wrong. Have a look at the first half of the sentence again, and you will see the structure clearly.

Yes, in this case you can remove the "to" and the sentence would make sense. In fact, it wouldn't make sense with that "to" in there. Since you have the "have to" you don't need that second "to". For that second part, no you cannot say "All you have to do is pressing the button" because "pressing the button" is something that's happening in the present. It would make sense if you said "All you have to do is continue pressing the button" which would essentially mean you're telling the person to continue performing an action they're currently performing. Hope this helps!:)