How many times can I tap the different vases until they become a lower level again?

Level 7 -1

Level 8 -2

Level 9 -5 (from response in this forum)

Level 10 -10 (from response in this forum)

If merging level 8 or 9, you can tap one less than max and it keeps level. So tap each L8 once before merging into a L9. Tap L9 4x before merging. And if you are going to trade in the L10 for an order (yeah, I had to do it), tap 9x first.