Answer 1: NEXT — 4 letters
Trace NEXT through 1 at row 1, column 1; 2 at row 1, column 2; 3 at row 2, column 2; 4 at row 2, column 1. The letters read N → E → X → T, so every step changes exactly one row or one column.
The movement sequence is right, down, then left. It makes two turns across three straight segments and stays within two rows and two columns. The route visits an edge three times and a corner once. After these four cells are assigned, 26 open cells remain for the longer answers.