2. The best design for five pieces using cuts that are parallel to the edges of the original square is shown in this figure: This gives a total perimeter of 4(2 + 1/sqrt(5)), which is about 9.8.
3. For nine pieces without the child-cut rule, we get this beautiful design: It gives a perimeter of about 15.3.
4. The best design for five pieces without the child-cut rule that permits cuts at an angle to the original edges is shown here The slant is about 19 degrees from the horizontal. This gives a total perimeter of under 9.4.
T.J. Takei provided these solutions. For the tilting calculation, he used the free software Gnu Octave.