It is also called an arrowhead. It is a kite with one of the vertices between equal sides "pushed in" so that it becomes a reflex angle.
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No. A trapezoid has two parallel sides. This is not possible in a concave quadrilateral.
It would be a concave quadrilateral The link below has some pics.
Yes. All polygons with 4 or more sides can be concave. An arrowhead is an example of a concave quadrilateral. The back of an envelop (where the sides are folded and glued together) is a concave pentagon.
A trapezoid (UK trapezium) is a quadrilateral has only one pair of parallel sides. Otherwise, there are many irregular quadrilaterals that have no parallel sides, such as a kite or dart shape, and these include both concave and convex quadrilaterals.