Can a square be a kite or a trapezoid and can a trapezoid or kite be a square? I don't think so, but this question is debated everywhere. The only reason a square is considered a rectangle and rhombus is because it is essentially a combination of the two. A trapezoid and kite had nothing to do with the creation of the square, therefore I don't think a square can be a kite/trapezoid and vice versa.
Your answer greatly depends on what definitions you are using.
For example, according to Wikipedia,
Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition) [...]. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition) [...].
So, for example, if you use the exclusive definition of a trapezoid, then a square is not a trapezoid. But if you use the inclusive definition of a trapezoid, then a square is a trapezoid.
The situation is the same for kites. You'll need to specify your definition before you can answer your question.
Edit: You state in a comment that you're using the exclusive definition of a trapezoid. So,
A square is not a trapezoid.
Now, I assume that by kite you mean "a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other," then
A square is a kite.
Using the definition of a square,
A trapezoid is not a square.
A kite is not necessarily a square.