For any convex quadrilateral, the area is 1/2 the magnitude of the cross product of the diagonals. In this case $${225\over 2}|\sin(\theta)|$$
Here $\theta$ is the angle between the diagonals. This is obviously maximized when $\theta=\pi/2$. Note there are many different isosceles trapezoids with equal diagonals of 15 that attain this maximum area.
Next, consider the rectangle with vertices $A=(x_0,y_0)=(7.5\cos(\theta),7.5\sin(\theta))$, $B=(x_0,-y_0)$, $C=(-x_0,-y_0)$ and $D=(-x_0,y_0)$. The perimeter is then $p=30\cos(\theta)+4\sin(\theta)$. Thus $p$ can be arbitrarily close to 30 by choosing $\theta$ to be sufficiently close to 0. I believe, but can not prove, that any isosceles trapezoid with diagonals of 15 has perimeter at least 30, but no such trapezoid attains the minimum of 30.
Edit:
I feel a little foolish. This was definitely a case of the forest hiding the trees. In triangle $ABD$, $|AB|+|DA|>|BD|=15$. Similarly for the other two sides of the trapezoid. So the perimeter is strictly greater than 30.