But you have another issue: Even if you are just below the speed of light, most of the mass would become relativistic mass (and relativistic momentum), with almost no rest mass.
But there's a postulate that only rest mass can make a black hole, and relativistic mass doesn't count. (Because otherwise you could travel fast and see inside the black hole.)
So we are left with a contradiction.
As for the effect of moving masses, that kind of "drags" space-time along with it. And rotating masses do as well. In the case of rotating masses that's called https://en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precess... and was what Gravity Probe B was measuring.
As for the light, you can solve the mystery as follows. Go to the reference frame in which the light is just all pouring into one spot. In that spot photons are meeting photons and creating a sea of pairs of particles - and now it is obvious that a black hole could be made.
So at that meeting spot they would go right through each other.
When a particle meets its antiparticle, you get 2 energetic photons. All such interactions are perfectly reversible. So when 2 energetic photons meet, you can get a particle and antiparticle. Both energy and momentum are conserved in this process.
But here we explicitly placed the photons to be traveling in the same direction, so there is some net momentum that can not be carried by the particles.
i.e. particle anti-particle annihilation will never produce two photons that are both traveling straight forward. So the event is not reversible.
You can read more about this at https://en.wikipedia.org/wiki/Two-photon_physics.
I read about the two photon physics - and the two photons can scatter off of each other, but they can't produce particles except in very high electromagnetic fields, which isn't the case here.
The process γ γ′ → e+ e− is a classic calculation in QED, and although it's never been observed directly for two single/isolated photons[0], there is no good reason to believe QED should be wrong in this particular case and, indeed, all evidence we have is pointing towards this process being perfectly possible[1].
[0]: https://en.m.wikipedia.org/wiki/Breit%E2%80%93Wheeler_proces...
[1]: https://www.energy.gov/science/np/articles/making-matter-col...
That's news to me. Could you provide a reference? "Relativistic mass" is just rest mass + kinetic energy, and so yes, it should influence the energy-momentum tensor like everything else.
> (Because otherwise you could travel fast and see inside the black hole.)
How so?
To see that this is a necessity, even for a single particle moving at relativistic speeds, consider energy conservation: If you accelerate a particle, you have to expend energy and that energy needs to come from somewhere (say nuclear fission). As a consequence, this energy will already show up in the energy-momentum tensor before acceleration (in the fission case: as rest mass of your isotopes) and, by local energy-momentum conservation (= 4-divergence of the tensor being zero), it then must also show up in the tensor afterwards.
Whether or not that energy can lead to a black hole is a different question. For a single massive object flying at relativistic speeds in an otherwise empty universe: Probably not, because one can simply transform that "relativistic mass" away by switching coordinates, as you say. For several objects moving relative to each other at high speeds, or even a single particle moving relative to the "rest of the universe", it's not so simple, though – you won't find a coordinate system in which all terms in the energy-momentum tensor are small (let alone zero) everywhere.