No, some kites fly badly with a tail. The tail add stability to a badly balanced kite but it also adds weight so can drag a gently pulling kite down. Unless a kite is homemade or asymetric in design it should be well balanced. As most kites are now mass produced and quite accurate people mostly use them to look pretty! A few kites (Sodi's) have rope tails to add weight at the base and produce the correct angle of attact to the wind. A looped tail made from a single piece of fabric and attached at two points on the base of the kite is used on some kites such as smaller sleds to add to the lift.
Kites such as box kites and tetrahedral kites are stable without tails. The most common kite requiring a tail is the flat diamond kite.
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All kites are quadrilaterals, but quadrilaterals are not kites.
no
Not all trapezoids are kites. A trapezoid is a 4-sided figure with a pair of parallel sides, so any item, not just kites, can manifest this shape.
Yes, because it has 4 sides
Yes. Any rhombus is also a kite (but not the other way around), and since any square is also a rhombus, any square is also a kite.