Indeed it is related to a standing wave, see solution below. The solutions are parameterized by the number of nodes.

There is an analytic solution in a closed form, see my solution below.

but the "gravitating mass" does increase with velocity in general relativity. In other words if you fill a box with gas and heat it to relativistic velocities, it will gravitationally attract a test particle outside of the box more, doesn't it?

Agreed, the center of mass velocity of two photons is timelike, so in the center-of-mass frame they are literally at rest and their invariant mass is their total energy. But imagine a spherical shell expanding isotropically with the speed of light. The enclosed mass is zero once inside of the expanding shell of radiation. So while the invariant mass is finite for a system of photons, the gravity they generate at infinite time must always be exactly zero.

Hmm, the last sentence sounds strange. Isn't the invariant mass of photos always zero in vacuum? If not, is their velocity timelike? Also, since the interactions propagate with the speed of light, they will never catch up with a photon, so a system of two photons cannot interact except when they collide.

If you have negative interference you do not have any energy flux left for detector 2 to receive.

The OP did not ask about a drain nor a plane wave. You can have two wavepackets shifted in phase by pi, and you guide the two wave packets into the same region so that they cancel. This exists in practice, e.g. the laser beam in the LIGO cavities, two beams are combined to have negative interference.

A monochromatic electromagnetic wave is described by a vector that specifies its wavenumber and propagation direction, and 2 more numbers: amplitude and phase for both polarization. In order to have negative interference between two waves all of these must exactly match except for the phase, which must be shifted by pi. If the other quantities do not match you will NOT have complete cancellation. None of the examples in this answer are relevant to the question as they do not produce complete negative interference.

How do you define the indices of the coordinates in your answer? Does the time index correspond to "1" in your result or "4"? (The usual convention is t=0, r=1, theta=2, phi=3, but your index range is between 1 and 4.)