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Oh, dude, you're asking about a kite! Yeah, a kite doesn't have rotational symmetry and its diagonals are not perpendicular. It's like that one shape that's just doing its own thing, not conforming to the norms of the quadrilateral world.
Yes. Any equilateral shape can have both rotational and line symmetry.
A square is on example. The perpendicular bisectors of the sides and the two diagonals comprie four lines of symmetry.
A semicircle.
The diagonals of rectangles are rotational lines of symmetry but not reflective. To be reflective lines, folding along the line has to give the same shape on each side.