Very interesting question, but also kind of mislead, which isn't immediately obvious.
So let me get one thing out of the way first: In terms of observation we are hard stretched to find 'just' empty space at all. Even absolute vacuum has an energy density associated with it, which is believed to be a result of quantum fluctuations, or spontaneous emergence of particle pairs from literally nothing (when explained in terms of a particle universe). This makes way more sense mathematically and intuitively when you think of the universe not as a volume containing lots of particles interacting, but when thinking of it as a collection of fields and their interactions. Any particular point in space, empty or not, has a field value associated with it, and the particles are just their 'interaction devices', or messengers, if you will.
You will have heard of the Higgs-particle which is said to give mass to electrons and protons an so forth. But actually it is the interactions of these particles with the Higgs-field giving them mass, the Higgs boson only being a measurable fluctuation of the field, if you will.
So that was one of your questions: What happens if I am the only object in the universe? Well, if the Higgs field is intact, physics will work as we know it. If not, space would be a very different thing and you would just move at the speed of light, because the absence of the Higgs field would essentially make you a photon. You could only move at the speed of light.
Of course, if you were alone, you would have to cut yourself in two pieces in order to know that there is gravitational interaction and hence, a Higgs field. But other than your ability to gain knowledge about the world, anything else would be unaffected.
You used the graviton as an example (which to my knowledge is still hypothetical, but that doesn't matter). Keep in mind that gravitational interaction is just concerning gravity, not space itself: It is the mediator of gravitational interaction, bound by the speed of light - in other words, its speed is dependent on spacetime, not the other way around (the speed of light being defined by it).
So let's just think of spacetime as the dimensions for our universe of fields:
Newton thought of them as absolute in the sense that nothing can influence them and everything is defined by them. He even kind of dismissed notions of it being relative, which made sense at the time: When a measurement is dependent on the observers location, speed or point in time, no theory would ever make much sense. He wanted physics to be about an objective frame of reference, in which there was agreement about the laws of nature. Copernicus' view of the solar system, for example, was just that in a sense, a theory which didn't make the observer 'special'. But that was also at a time when people thought of causality of being instantaneous, or the speed of light being infinite.
What Einstein did with that was to turn it around in a way. If the observer mattered for every measurement, he concluded that space and time must be influenced by the things going on. Now, whether spacetime really is bent and curved and so on really does not matter to the theory: You might as well say that everything just behaves as though space is curved, but the outcome would stay the same. So in other words, we do not really have a way to decide the question of wether space really behaves as Einstein thinks.
Maybe keep these things in mind when you move on:
- In any field of physics, or other sciences, space and time are dimensions in which other things are defined. So, 'stuff' or energy or whatever, really needs spacetime in order for us to make any sense of it. The only exception where space and time itself is even a 'thing' to be influenced is the theory of relativity, but that may well be because we do not understand the quantum nature of relativity yet.
- Think of the speed of light as the speed of causal interaction. Many questions related to this (Like for example the edge of the universe) makes much more sense this way.
- Modern physics is best understood when thinking of nature as the product of fields and their interaction, not of particles in empty space
I hope this helps!
