First answer: GAIN
Trace GAIN through 1 at row 7, column 1; 2 at row 7, column 2; 3 at row 6, column 2; 4 at row 6, column 3. The cells spell G → A → I → N in that order.
Move right, then up, then right. The direction changes after both the first and second moves, giving this short trace two turns and three straight segments. It spans two rows and three columns, visits two edge cells, and includes one corner. Its starting cell has one open orthogonal neighbor, while its ending cell has three. Assigning these four cells leaves 29 open cells for the remaining routes.