Confirmed by the clarifications: Tennin Institute can place advancement tokens on identities and installed Runner cards.:Ruling
Tennin Institute's ability does not trigger on the first turn of the game as the Runner has not had a "last turn" yet.:Ruling
If the only run on the Runner's last turn was against a server with a Crisium Grid installed in it, Tennin Institute gets to place an advancement token on a card, even if the Crisium Grid was trashed during that run.:Ruling
On the first turn - does Tennin Institute fire? Tennin Institute does not work on the first turn, since the Runner did not have a turn. You cannot Trick of Light off of Tennin Institute, since Trick of Light only works on installed cards and IDs are not installed.
Corp is playing Tennin Institute. Runner's only run in a turn is against a server with Crisium Grid on it. Run succeeds, Runner accesses & trashes Crisium Grid.
Corp turn starts - do they get to place an advancement counter from the Tennin ID? You are correct. The Crisium Grid would allow the Tennin Institute to place an advancement token, since the run was not considered successful for the purposes of all card effects.