Not necessarily, in fact, only parallelograms have them, irregular quadrilaterals do not.
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Yes, at least one
An arrowhead is also called a delta. It is a concave quadrilateral with one axis of symmetry.
The shape described is a kite, which is a quadrilateral with two pairs of equal adjacent sides and one reflex angle. A kite has an axis of symmetry along the line connecting the midpoints of the non-congruent sides. The reflex angle in a kite is typically greater than 180 degrees, creating a distinct shape that is not a regular quadrilateral.
A kite or an equilateral trapezium.
An axis of symmetry is any of one or more lines about which a geometric figure is symmetrical.