What is the lowest level things have temperatures? Something can be cold. If you split it in half, it will still be cold, keep doing that and it will still be cold. My question is, what is the lowest level things can have temperature? Do atoms have temperatures? Do quarks?
 A: Any "system/object" can be in a thermal state and thus will be well described by concepts such as temperature.
Talking about things in thermodynamic terms such as temperature express a degree of personal ignorance about the system, in which case you talk about an average or expectation of certain general properties (such as overall energy etc).
As an example of arguable the simplest fundamental object, consider a qbit. This is an object that for some measurement will only have two possible outcomes (for instance it could be a spin-1/2 particle in which case it can be measured as "spin up" or "spin down"). Now if this object is put in contact with a thermal bath/environment that is at temperature $T$, after a certain amount of time the qbit will also be in a thermal state at temperature $T$ even if the qbit is then decoupled/removed from the environment! Physically this means that if you make a measurement you will find the qbit in state 1 with probability $P_1$ or in state two with probability $P_2$ where the probabilities are those given for a thermal state at temperature $T$, i.e.
$$\frac{P_2}{P_1} = \exp\left(\frac{E_1-E_2}{kT}\right).$$
A: A small system coupled to a temperature bath will have a well defined temperature (the temperature of the temperature bath), if you do not disturb it out of equilibrium.
If a single, isolated particle is considered, things are not that tricky either. You can always consistently assume a temperature of zero (but you gain nothing by doing so).
But a small collection of isolated particles is the case where it gets tricky, you will not have a well defined temperature!
It is even possible that a macroscopic system does not have a well defined temperature, provided it is driven sufficiently strong and therefore cannot reach equilibrium (or a steady state sufficiently close to equilibrium).
From another point of view, describing a system's properties by its temperature is just a trick to make difficult things more easily. Instead of considering all the microscopic interactions with a bath, you assume thermal equilibrium and therefore get simple relations certain quantities. This point of view is especially natural when considering the problem from a stochastical processes perspective (where the interactions with the surrounding medium are simply modelled by delta correlated noise with a strength dependent on temperature).
All in all, there is no definite answer to your question. A small system coupled to a thermal bath, will have a well defined temperature, in this sense an atom can have a well defined temperature.
On the other hand the concept of temperature looses its sense, when one moves away from the thermodynamic equilibrium, so in a way no real system actually has a temperature.
While it is difficult to assign a temperature to small systems, this does not matter when the system is isolated, as once you consider all microscopic interactions there is no need to consider the temperature (as handling temperature is, as mentioned above, in a way just a hack to simplify calculations).
Put yet another way, temperature is the property of an ensemble, not of a single system. A large system can always be considered an ensemble and therefore be assigned a temperature in that way. For a small system it depends on the way the system is prepared. In a canonical or grand-canonical preparation it is fair to say the system has the temperature of the bath, even if you disturb the system out of equilibrium. For a micro-canonical preparation of a small system assigning a temperature does not make too much sense, but the quantity is nevertheless well-defined (but utterly irrelevant for a system instead of an ensemble).
A: Temperature is a measure of mean kinetic energy. So as long as particles can move, they will have temperature.
And yes, quarks do have temperature. The highest man-made temperature was attained in a quark-gluon plasma at LHC in 2012.
A: One definition of temperature is that it is the parameter which determines the distribution of velocities of an ensemble of particles. Note that I refer to an ensemble (a group) of particles. If you truly continue breaking down into smaller and smaller pieces, you no longer have the concept of a distribution and things become tricky.
You can actually define the temperature of only a few particles by looking at only a handful of fermions, and asking how long do they typically occupy their single/doubly occupied states. Quantum dots, only a few nanometres in size typically, can have a well defined temperature assigned to them. See Wikipedia for a bit more detail.
All this being said, temperature is a very weird concept, and I'm not sure that it is well understood in the extremes.
