Preveina from Crest Girls' Academy sent us some pictures to support her reasoning about some of the shapes in this problem:

A circle can be always fitted in a square touching all 4 sides since the sides of a square are all equal. This makes the circle touch each side of the square evenly.

A circle can never be fitted in to a rectangle touching all 4 sides because a rectangle has 2 long sides and 2 short sides. When you're trying to draw a circle that touches all 4 sides in a rectangle it'll turn out to be an oval, since there are 2 long sides.

A circle can never be fitted in to a
parallelogram touching all 4 sides because a parallelogram has 2
long sides and 2 short sides just like a rectangle has.

A circle can also be fitted in to a
kite touching all 4 sides.

The circle can be sometimes fitted
into the trapezium touching all 4 sides depending on the length of
the sides. If the non-parallel sides are too far apart, the circle
becomes stretched into an oval.