Yep, that's what I did and printed my Covid
Would not have thought so. Fully vaccinated is currently defined as 2 jabs and then you get your COVID pass.
If they change the requirement to two jabs and a booster (or a 3rd jab for the vulnerable), then as long as you have had that, then your COVID pass will still be valid. If not, your pass will no longer be valid.
I obtained the covid pass letter from NHS online, 4 weeks after I had my booster on 8th November. The booster does not show on the letter; I have asked for another and hope that will show it; if not I shall have to asked someone with decent Techie ability to install the app for me. I'm in the "Broken Tonka Toy" group!