AskPhysics

Not 100% certain but I think the answer is yes. In general relativity the main equations revolve around two things: the metric tensor and the stress energy tensor. And of course we all know that a tensor is the thing that acts like a tensor (j/k but I probably can't explain properly, lol).

The metric tensor is (basically) describing the gravitational potential across spacetime, so the mechanism of 'pulling' a particle. And the stress energy tensor has the information about the energy and momentum flux in the region, which is what affects the structure of spacetime (ie the metric). So mass creates this flux and we get normal everyday boring gravity and also photons create an energy flux, and make some kind of funky but functionally identical but very weak gravity.

It wouldn't really be noticeable in practice. I would be interested to see the maths done but don't want to spend Sunday working out tensor components.

For the second part I don't know, in a way? If a region has more stars moving through it, leaving and then some time after another one coming, does that region have more gravity? In a way I guess

One could answer "yes" to your title question. To your text questions the answer is undefined, because the questions themselves start from somewhat incorrect statements.

I say "could/would" because it's always tricky to interpret someone else's questions: maybe you have some intuitive pictures in mind, and I may mistranslate them. So take the following with a pinch of salt!

Let's take the viewpoint of General Relativity. Then "gravitational pull" is really the curvature of spacetime, that makes bodies and radiation gets closer or go apart. According to general relativity, the curvature at a point in space and time is generated by (one could almost say "is the same as") the amount of energy, energy flow, and pressure (more exactly "stress") at that point.

Now photons, or more exactly electromagnetic fields, always have an associated energy, energy flow, and pressure. So their energy-force generates curvature.

The equations of general relativity also say that curvature at one point in space and time influences curvature at other points in space and time. So, in principle, a photon passing through a region of space is contributing, owing to its energy-pressure, to the curvature in that region, and therefore to the curvature in nearby regions, which can be experienced as gravitational pull.

But it's also good to keep in mind that the photon's contribution to curvature may be extremely small, depending on its energy-pressure.

Regarding your first text question, it isn't clear what you mean by "massive" region. It's actually a tricky matter to speak of the "total energy" or "total mass" of a space region – it's basically a badly defined concept: different people can arrive at different numbers because they use different conventions. There's a nice discussion about this at this link. Besides that, one must also be very precise in defining "a region of space". Because what you call a 3D region of space may for another observer be a 2D region (say a surface) moving in time. So to be honest I prefer not to try to overguess your question and give you an answer.

There are many books where you can find this info. A very accessible and publicly available one is General Relativity by Crowell. The standard reference in these matters is Gravitation by Misner, Thorne, Wheeler.