Yes
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No. An isosceles trapezium (isosceles triangle with its apex removed) would have congruent diagonals but it is not a parallelogram.
Yes. Any triangle will tessellate.
isosceles triangle
It depends. Strictly speaking, a semi-regular tessellation uses two (or more) regular polygons and, since neither an isosceles triangle nor a parallelogram is regular, it cannot be a semi-regular tessellation. However, a less strict definition allows non-regular components.