2) what does it mean that a photon has zero rest mass but non-zero mass “while moving”? I’ve seen calculations that show light beams attracting each other in some cases (IIRC parallel light beams remain parallel, but “anti-parallel” beams always converge), and I also saw calculations of black holes formed by infalling shells of radiation rather than matter.

A moving photon does not have nonzero mass, it has nonzero momentum. In the Newtonian approximation we calculate momentum as p=mv, but this does not work for photons, where we instead use the full relativistic equation E^2 = m^2c^4 + p^2c^2 (observe that when p is small compared to m, this simplifies to a rather more well-known equation), which, taking m=0, gives p = E/c.

As for light beam attracting each other, that's an electromagnetic effect described by high-order Feynmann diagrams, like the one shown here. (At least, that's true if I'm thinking of the same calculations you are.)

1) charged black holes are studied separately, and those solutions seem to look different than non-charged black holes?

3) doesn’t energy-matter equivalence imply that fields that store energy should bend space like matter does?

Both good points. I'm afraid we're a bit beyond my expertise; I'm now unsure even about the electromagnetic Tipler cylinder.