Did the player have to be inbound to score a touchdown?

Answer 1

Yes.

The technical way to score a touchdown is to have to ball "cross the plane" into the endzone without the player being down or out of bounds.

With a catch, the player must establish himself as inbounds while maintaining possession. In the NFL, the player must do so with two feet inbounds. In NCAA Division I football, the player only needs to do so with one foot inbounds.

With a run, the player must have not stepped out of bounds before the ball crosses the plane into he endzone.