Trapezoids do not need to have any angles that are equal to one another is correct because it is possible for a trapezoid to have 4 different angles.
A trapezoid does not always have a right angle. A right trapezoid is a trapezoid that has one right angle, but not all trapezoids have a right angle.
At least two angles in a trapezoid must be supplementary, which means that they add up to 180 degrees. However, it is possible for a trapezoid to have more than two angles that are equal.
The perimeter of a trapezoid is equal to the sum of the lengths of all four sides. The formula you provided is for the area of a right trapezoid, not the perimeter.