BTAs can and will climb glass walls if they're unhappy. Assuming that you've got enough light and good water quality, etc., it is best to make sure that the rocks the anemone is on aren't touching the glass.
Even so, if it's unhappy enough, it can release its foot and float in the current seeking a better place. Unfortunately, that can end badly. The good news is that they don't usually do that unless things are so bad that it can't get what it needs where it is.
As to the mushrooms, the nem may kill them, or the mushrooms may irritate the anemone, causing it to move.