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Yes, any type of triangle or quadrilateral will tessellate.
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, any quadrilateral will tessellate.
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.