Inertia of a box of photons Suppose a box one light year long (along its x-axis) with some mass M also contained a bunch of high energy photons with energy E that were travelling along the x-axis.  Would the inertia of the box include the energy of the photons while they are travelling freely along the axis (not bouncing off the walls)?
 A: The problem with your question is that you're trying to describe inertia in terms of what are, inherently, transient phenomena. Never mind the photons, if you push on a box that large it is going to take a long time for the sound waves to propagate from where you pushed to the rest of the box. So, for the specific limits you've given, you'll find an inertia that is on the order of $$\frac{v_s}{c} L \mu,$$ where $v_s$ is the speed of sound through the walls of the box, $L$ is the length of the box, and $\mu$ is the mass per unit length of the box.
If you really want to measure the inertia of something you have to apply a force for long enough for internal wiggles and jiggles to damp out, including photons bouncing back and forth. Otherwise, you're not really measuring the inertia of the whole object, but the fraction you've been able to affect, with errors for the complicated movements a non-perfectly rigid body can make; and relativity rules out all perfectly rigid bodies because they would permit sound to travel through them faster than light.
Here's a slinky drop demo of how the finite speed of sound through an object can affect that object's dynamics in interesting ways: Veritasium slinky drop.
A: When you prepare at t=0 a localized electromagnetic pulse wave containing you photons then the movement of the walls of the box (that are 1/2 light year away) and thus its inertia should not be influenced by the wave pulse (and thus photons) inside. The reason is that no electromagnetic waves would be reflected at the box walls which is necessary for a momentum transfer. On the other hand a small size cavity with standing electromagnetic waves would have an inertial mass contribution corresponding to the energy of the electromagnetic waves (photons) which has been pointed out in 1904 by Friedrich Hasenöhrl.  
