Counter in Excel

All of your requests can be accommodated. But only using VBA I'm afraid.

Keep the dropdown but also add a button that triggers a macro.

In the macro, read the cell with the dropdown and send the appropriate sheet to print. Then update the appropriate counter which can be in a set of cells on the same page as the dropdown and button.

To count the number of saves, you need a macro that before_save function. You need to search for that, I think there may even be an article about it on my blog, can't remember, it's been a few years since I've done it. The principal is the same as the previous macro in that you simply add 1 to the appropriate cell.