Does a box containing photons have more inertia than an empty box?

Yes! In fact, this is very common. For example, the mass of a proton is much greater than the sum of the masses of the constituent quarks. Much of the extra mass comes from the gluons that bind the quarks together; each gluon is massless, but collectively they contribute to the inertia.

The point is that the mass of a system is not the same as the sum of the masses of its constituents. Of course, this is just a rephrasing of $E = mc^2$. If you have photons bouncing back and forth in a box, their energy contributes to the total mass.

Yes, mass and energy are equivalent. A more competent relativist might be able to give you the complete description, but to first order you can say that the mass of an object is simply the total energy in its volume divided by c^2. That mass is equivalent to the inertial mass by the weak equivalence principle, which is a cornerstone of GR.

That is to say, the answer is yes by the weak equivalence principle.

Yes, both the internal potential energy and the internal kinetic energy of a bound system (in the rest frame of its center of mass) contribute to the bound system's inertial mass according to $E=mc^2$. For a paper discussing the evidence that this is true for internal kinetic energy in particular, see Kinetic Energy and the Equivalence Principle.