With our P&O cruises, I up-loaded everything that was required onto the VeriFly app (after a lot of difficulty I might add!). At check-in we had to show paper copies of our vaccination status. Apparently the check-in staff do not have access to the VeriFly information.
On board we never use our phone so we did not use the app at all. When you get to your cabin there will be a letter that lists the dining bookings that they have (kindly) made for you. You can change these, but it is better to do it as soon as you get to your cabin, otherwise the time slot you want in your preferred restaurant might not be available.
Forget what happened last year - menus rotating every three days in the Waterfront. We were on a ten-night cruise in April/May. The menus (breakfast, lunch and dinner) stayed the same, but with one item changed in what was listed as "Crew Favourites". I think the menus were shown outside the restaurant, but if you want a popular time, you wouldn't be able to get in without having a booking. If you want to dine late (around 8pm) you shouldn't have any problem.
We didn't go to Dragon Lady, but in Angelos, the menu stayed the same from what I could see with our three visits.
I don't think you would have a problem requesting a table for two most of the time.