Is it possible for an equilibrium to be Trembling Hand Perfect for one player but not another in an asymmetric game? Is it possible for an equilibrium to be Trembling Hand Perfect for one player but not another (in an asymmetric game)?
Yes. It is possible for a strategy profile of one player to be trembling hand perfect and not the other.

Generally speaking, trembling hand perfect is used in the equilibrium sense. Certainly for an equilibrium to be called trembling hand perfect, the strategy profiles of all players must also be so.