Would time exist in a universe with only one elementary particle in it? Imagine a universe infinite in all directions with only one elementary particle moving through it (impossible, but suppose). Can one define time in this universe?
 A: No, you cannot devise a meaningful definition of time in a single-particle universe. Even defining XYZ space meaningfully becomes highly problematicin that scenario. 
Any meaningful definition of classical time requires the existence of entropy, or more specifically of the ability to create classical information through selection of some subset of multiple possible configurations. Since no such set of multiple configuration is possible with a single particle in a stable state, that particle necessarily remains unchanged and so timeless.
An even more interesting question is this: What is the minimum number of particles required to make time possible? It's trickier than it sounds because classical physics very often assume the existence of a godlike external observer, one capable both of recognizing cyclic behaviors (clock ticks) and of recording information (the observer is subject to entropy).
Both are flagrant forms of cheating, since they simply borrow the time flow of domestic very complex entropy observer and apply it to a much simpler system.
So, the real question is this: What is the absolute minimum of complexity within a small universe that enables meaningful definition and measurement of time?
This question turns out to be much more closely akin to Turing's question of what is the absolute minimum set of components needed to create a computing machine with memory. The minimal time universe will for example require both some form of recognizably cyclic behavior (clock ticks) and a multiplicity of possible states that can be reduced (memory). The simplest such universes will only exhibit time briefly before they reach entropy death and cease to change. 
A: Well, time is a dimension defined independently of energy/matter. So, yes.
In fact, we frequently talk about (mathematically) universes that are completely empty; have no matter whatsoever (and sometimes no energy either, but that's a tough thing to define). Those spacetimes we define have time still. It's a part of spacetime; it exists; regardless what populates spacetime.
Time is a dimension, which means it is used to specify the coordinates of an event in spacetime. You might say that an empty universe would be unchanging in time and that if every point in time is identical, there's no way to tell how much time passes and, thus, time is essentially not really there. I say even if this empty universe is unchanging through time, it still has that dimension. The mere fact that everything is exactly the same at every moment of time merely means there is additional symmetries in this universe (and additionally fun conservation laws that will ultimately end up affecting nothing). You may not be able to tell exactly how much time passes, but you know that time does pass. One moment is not the same moment as the next (it looks exactly the same, but if it were the same moment, then it wouldn't be the next moment).
The next thing you might say is "but the spacetime interval is only useful for describing the separation between events and if there is nothing around to take part in an event, how can you talk about the spacetime interval meaningfully?". That's an excellent question. A universe without even one test particle in it is not really worth discussing. But when forced to, it's easy to say "there is time because we can still define the universe with time". Of course, you could define a universe with no time or with an arbitrary composition of dimensions, but defining something without a dimension just to show that it doesn't have that dimension is really cheating. That's like saying "yeah, well in a hypothetical world where birds don't exist, birds don't exist. Therefore, there could be a world where birds don't exist". See, it's cheating.
A: For fundamental questions it might be helpful to consider primarily proper time instead of coordinate time because proper time is the more fundamental concept.
You will find that a massless particle has no proper time. If there are only massless particles in the universe, that implies that there are no observers which could observe the massless particles moving at speed of light within a certain time lapse. Massless particles would not generate any time.
The worldline of mass particles is generating proper time. If there is one mass particle in the universe, this particle would be subject to some form of aging, the spin and the wave of the particle would have some frequency and some phase which may be counted in the same way as clocks are counting time. That means that one mass particle would be sufficient for the existence of time.
But you can also think of time in a universe without any particles at all. Space expansion is a process which is working like a clock. 
A: In a (non expanding) universe with only one elementary particle (with a fixed four momentum) in it, time and space are without measure and symmetric. In the context of the weak force there is a non-symmetry of C(charge) P(parity), compensated for by a non-symmetry of time, but that's of no importance here.
With some extra particles put in the universe you can give a measure to both space and time. For time you can use the frequency with which an outer electron revolves around the nucleus in a stable atom.
For the measure of space you can put two hydrogen atoms at a fixed distance and make that distance the measure. Or maybe you can use the diameter of the particle itself (I truly belief an elementary particle has as non-point like structure with a diameter near the Planck-length).
But space and time will still be symmetric. To say time is symmetric seems to me the same to say there's no time at all. This symmetry is broken by the introduction of a great amount of particles for the concept of entropy to emerge.
