A pyramid
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Quadrilaterals are polygons with four sides. Triangles are polygons with three sides.
A shape formed by four triangles would have to be a tetrahedron. But I believe that a tetrahedron can have at most three right angled triangles. One with four of them is, I think, impossible.
No. Triangles have three angles and three sides. A quadrilateral has four angles and four sides.
No. Quadrilateral means "four sides." Triangles have three sides.
Because the sum of the sides of three triangles is nine... Placing three triangles together so they all touch effectively removes four of the total number of sides - leaving five.