Given one side, the opposite angle and the area of the triangle, is it possible to solve for the other two sides?

Answer 1

The inscribed angle theorem prescribes the possible locations of the opposite vertex - once you find one triangle with the correct segment and opposite angle, all possible locations for the opposite vertex lie on the circle containing the vertex you found and the two endpoints of the segment.

Answer 2

If it is a right triangle, then you have a chance.