Yes, they are because it would just be a scalene triangle if they weren't.
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If you mean an isosceles triangle then no because all triangles have no diagonals
You can't because triangles do not have diagonals but an isosceles triangle has 2 equal sides
No. An isosceles trapezium (isosceles triangle with its apex removed) would have congruent diagonals but it is not a parallelogram.
A trapezoid with congruent diagonals is an isosceles trapezoid.
Only the diagonals of isosceles trapezoids are congruent.