A couple of ways of looking at this. The first one is that gravity is self interacting -- the equations of General Relativity are nonlinear, and obtaining solutions involves having all the higher order effects taken into account. Any matter energy can interact with any other matter energy gravitationally. In a posible quantum description of gravity (which we still don't know how to so, but in the simplest semi linearized view) even the gravitationa quanta, the gravitons, will interact with each other. It can get pretty complicated.
The other way is to imagine some cases where we have already obtained solutions, and examine whether they provide an answer. One example is black holes (BHs). As BHs are getting formed y collapsing exploded supernova the matter will be pulled into a tighter and tighter volume. As it does the gravity outside it keeps getting stronger. As the matter approaches the horizon, the gravity gets even stronger, and the time dilation (for an observer far away, as you asked), increases without limit and approaches infinity. Thus an observer far away sees the collapse slowing down, and never sees the matter going into the horizon. Another way of saying it is that a far away observer will never see the BH fully collapse inside its horizon. The BH collapsing causes the time dilation, and suffers it for an observer far away.
An observer falling in with the matter will not experience any time dilation, and after a finite time finds himself/herself inside the horizon, without even knowing he/she went through it. And can never escape. He/she, in a very short time, falls towards the singularity, gets very deformed and well, falls into the sigularity (or whatever quantum entity replaces it in a quantum theory of gravity).
Now, the large dilation happens, for an outside observer, much more strongly as the infalling matter approaches the horizon, where velocities are high and the actual collapse happens very fast for the infalling observer. For the far off observer, even with time dilation, it is still fast, and they would see the horizon being approached, and then sees nothing of the infalling matter or observer. So, even if it is an 'almost BH', there are no observable differences. The two BHs that merged and we observed it's gravitational radiation in 2015, we saw them merge in their last few orbits around each other in less than a second. That's why there are so many BHs in the universe, we believe, they are 'almost BHs', but for us ther is no observable difference.
So, yes, in any region in spacetime the gravitation is described by the curvature of the spacetime, and will affect anything there.
What of a single elementary particle, does its gravity affect it? We think it has to in some self consistent way to allow the particle to be what it is, but we still,have not developed an accepted theory of quantum gravity. When we try to use the normal rules of quantum theory to figure out those self effects, we get infinity. And unlike in quantum field theory where we have figured how to deal with those infinities to get finite results, we don't know how to do that in quantum gravity. It is what is called a non-renormalizable theory. So, we don't know the answer at the quantum level.