1. Any tournament with at least 7 vertices can be made strong by pushing at most one vertex (and there are at least two such vertices).
2. Given a tournament with at least 7 vertices. If as many as n-6 vertices are marked as "forbidden", you can push unforbidden vertices so as to make to make the tournament strong. (To see this, first note that the forbidden sub-tournament has a Hamiltonian path and focus on the two end vertices of that path).
3. All tournaments with 11 vertices can be made 2-connected using pushes. Thus the only open case is for 10 vertices.