Due to a leak, a horizontal tile from the top of our bathroom window fell. Here's a shot of the window from the inside with the missing tile at top center for context.

The leak seems to be fixed so I'm reattaching the tile. Would anyone be able to answer the following questions for me?
- I'd like to match the adhesive. Is this thin set, mastic or something else?
- Looking at the adjacent tiles that are still attached, the gap between sill and window is 1/2". Can the adhesive safely fill a gap that large?

- The fallen tile has most of the adhesive on it. Any ideas on how to remove it aside from gently going at it with a screwdriver and hammer?

Any tips on the above would be appreciated. Thanks in advance for you input.