Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one, depending on your background.) A pseudo-Anosov homeomorphism induces a fixed-point-free isomorphism of the Teichmüller space.

Edit: As Andy points out, any Dehn twist (about an essential simple closed curve) also gives an example.

Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one, depending on your background.) A pseudo-Anosov homeomorphism induces a fixed-point-free isomorphism of the Teichmüller space.

Edit: As Andy points out, any Dehn twist (about an essential simple closed curve) also gives an example.

Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one, depending on your background.) A pseudo-Anosov homeomorphism induces a fixed-point-free isomorphism of the Teichmüller space.

Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one, depending on your background.) A pseudo-Anosov homeomorphism induces a fixed-point-free isomorphism of the Teichmüller space.

Edit: As Andy points out, any Dehn twist (about an essential simple closed curve) also gives an example.