Can conduction alone cause thermal equilibrium? From the little I know about physics and conduction specifically, it seems to be that a system in which the heat transfer is done solely by conduction, cannot reach thermal equilibrium.
I'll preface this by stating what I think I know. Convection is the transfer of heat via atoms or molecules, meaning this is only present in fluids (and plasma I guess), as the atoms/molecules move more freely, as opposed to those in solids. Conduction is the transfer of heat via collisions of neighbouring atoms/molecules, and is thus more relevant in solids, as convection doesn't happen in solids and neighbouring atoms/molecules are further apart in fluids.
Now, it makes sense that convection would dissipate the thermal energy of a system, since the "heat-carriers" are actually moving around. With conduction though, this isn't the case. The heat carriers in a solid are to some degree stationary, and can only dissipate the energy to their neighbours.
From this minimal understanding, it seems to me that in an isolated system with conduction as the only possible method of heat transfer, thermal equilibrium cannot be reached. Instead, the atoms/molecules close to the heat source will be vibrating the strongest, and then their neighbours the second strongest, etc. I assume this is impossible, as I don't see how one could evade radiation. But, in this contrived scenario, with conduction as the only method of heat transfer, would the isolated system never reach thermal equilibrium?
 A: Given enough time, the thermal energies of neighbouring atoms/molecules will eventually become equal (or as close to equal as you like) and the system will reach thermal equilibrium. This just takes a longer time if the only available method of heat transfer is conduction (which is why oven gloves work).
A: In my judgment, your concept of convection is a bit flawed.  What happens with convection in fluids is that the conduction that is present in both fluids and solids (via collisions of molecules) is enhanced in the case of fluids by allowing colder regions of fluid to be brought into closer proximity to hotter regions of fluid (as a result of the fluid deformation and movement).  This effect enhances the temperature gradients, and thus increases the rate of conduction.  So convection is really flow enhanced conduction.
A: The three modes for heat transfer by conduction are molecular collisions (fluids), lattice vibrations (solids), and free electrons (free-electron solids such as metals and semiconductors).
The two modes for heat transfer by convection are natural (free) and forced. Convection can only occur in fluids. Convection can be modeled over infinitesimal distances as though it is molecular conduction. The addition over just pure molecular collision style conduction that the infinitesimal regions are moving relative to an external (static) reference frame.
Radiation transfers heat as black-body emission for the object at its temperature modulated by the emissivity for the object.
Thermal equilibrium occurs between two objects that are touching each other (e.g. between a system and its surroundings) when the two objects are at the same temperature. At that point, no net heat transfer occurs.
When two objects that touch each other are at different temperatures, heat transfer occurs $\dot{q}$ (W). The total rate for heat transfer (T) is a summation of the three modes: conduction (k), convection (h), and radiation (r).
$$ \dot{q}_T =  \dot{q}_k +  \dot{q}_h +  \dot{q}_r $$
The rate that any system reaches thermal equilibrium depends on the role that each term above plays in how that system exchanges heat with the object (or objects) in its surroundings. We can find systems that are equally rapid or slow to reach thermal equilibrium given any number of combinations in the above terms.
Finally, technically speaking, the terminology used in thermodynamics defines an isolated system as one that has no heat or mass flows. So, in answer to the last question, imagine an isolated system at a temperature $T_s$ surrounded by objects at different temperatures. Since no heat transfer occurs (the system is defined to be isolated), the system is not in thermal equilibrium with any object in its surroundings. When we contact this system thermally with any surrounding object, the system is no longer considered to be isolated. It may remain closed (no mass transfer occurs). At the point when the system is no longer isolated, it will strive to thermal equilibrium with the object(s) in contact at a rate controlled by the active modes for heat transfer.
A: I don't see what difference you think having radiation and/or convection would make. If we consider only two bodies, the heat transfer is proportional to the difference in temperature, and as heat flows between them, the temperature difference decreases. If $y$ is the difference in temperature, the derivative is proportional to the value: $y'(t) = -ay$ for some $a$. This give $y = ce^{-at}$ for some constant $c$. So theoretically, the temperature difference never reaches zero, but temperature is a stochastic property to begin with, and at some point the temperature difference becomes smaller than the fluctuations of molecular energy, so in that sense the temperatures are "equal".
The above is true for heat transfer in general (convection may introduce non-linear terms that complicate somewhat, but there's still an asymptotic decay to zero temperature difference). If you're okay with saying that radiation can reduce the temperature difference to zero, then I don't see how conduction is difference.
I did introduce the simplification of dealing with only two bodies, but you can model a solid object as being made up of infinitesimal regions, and then you have the same basic phenomenon causing the difference between two adjacent regions to go to zero. Then instead of having a one-dimensional function of time, a three-dimensional object would have a function whose input has one time dimension and three space, and whose output is one-dimensional (just temperature). However, the basic concept involved can be explored by considering just one spatial dimension, giving a function $y(x,t)$.
Your question is a bit inconsistent as to what situation you're imagining. At one point you say "isolated system", but at another point you say "heat source". If a system is isolated, that means there is no heat coming in (I suppose there could be heat generated by chemical reactions within the system, but those reactions would eventually run out), so you could have a portion of the object that starts out hotter than the rest, but after that, that portion just cools off. In that case, the system approaches a state where all of the object is the same temperature.
If there is a constant source of heat, then indeed the object won't reach a state where the temperature is constant across it. There will be some persistent heat gradient, and the object will reach a steady state where although the heat gradient varies across space, it's constant across time. Again, this is what will happen regardless of whether radiation exists.
A: I'd like to add that I think your thinking about the "isolated system with conduction as the only possible method of heat transfer" is somewhat confused by the "heat source" that you mention. To make reasonably physical assumptions, you'd have to assume the "source" contains a finite amount of energy, and starts at a finite temperature - it would probably be better to say this system has a "hot part" and a "cold part," rather than some abstract "heat source."
Heat would then conduct into the neighboring molecules of the hot part, which would then go into their neighbors, and then into their neighbors, and so on - heat would spread everywhere in the cold part. Meanwhile, the hot part is losing heat. While the hot part is still hotter than the cold part, heat continues to spread from it. Eventually, the hot part will have lost so much heat that it is now at the same temperature as the cold part, and heat transfers at the same rate into and out of that part, and the system has reached thermal equilibrium.
