The discount on board, if any, can vary wildly depending on demand and your negotiating skills. Generally, virtually everything in the planner, including specialty dining, is cheaper than on board.
You should be able to board at any time, but once the porters and terminal staff leave, it may get a bit delayed, and you'd have to haul your own luggage.
As seen above, there are activities for that age group. Ships were still coming back to full capacity and "normal" operation last June - that's why probably you didn't have any. The availability of activities also depends on the ship - more likely on the bigger ones.
The link is the same, but some weeks they don't post a new flier so the old one will come up under the link. Before last week, there was no flier for a month.
Set the app to the current sailing - that's about as current as you'll get in terms of a show listing. Usually, the same show is repeated through the week.