When two particles interact, do they spontaneously start treating each
other as particles in terms of potentials and so forth?
I'm not sure what you mean here. There are no particles at this level, just localized quanta that happen to follow certain rules. If you're talking about the interactions of something like QFT, then yes, but this is more a function of the field than particles... which of course becomes redundant.
Why we see these little things we call particles...I have heard that
we only see particles when we attempt to "measure position"
This is the Copenhagen versions (and others). Under Copenhagen there is a time-evolution of the wavefunction that, depending on what you're measuring, is becoming less well defined. Then a miracle occurs and you "collapse the wavefunction" and it takes on a particular eigenvalue. After a century we still have no cogent idea how this collapse is supposed to work (and a number of really dumb suggestions).
have a hard time wrapping my head around this in the MWI framework.
I think everyone does. As always, I suggest reading this for a reasonable introduction to the conceptual side of things.
In MWI, and the other no-collapse theories, the wavefunction does not collapse and there is no particular magic to a measurement. Measurements are anything that is thermodynamically irreversible, although I'm not sure if this is by design or an assumption for ease-of-use. I think this part of MWI is well-loved - it "makes sense" that you have one big universal wavefunction and everything is a relative state, and when you think about it, it's not clear how you would have anything else - definitions of "separate systems" tend to be non-physical hand-waving.
In any event, in MWI the weirdness moves from the collapse to the viewer. There is this infinite number of "yous" smeared out over probability space in precisely the same way "the particle" is. But for some reason we do not see that, we see this as a single person with a fixed notion of reality looking at a single particle. Of course this isn't deadly by itself, because everything sees that, so to answer your first question, the particles see other particles for sure.
But I'm not sure MWI has been truly successful in explaining why this seemingly simple mechanism of the universe is not what we see. The basic idea is that we don't really have an idea of reality anyway - visual illusions are a wonderful way to illustrate this - and what we think we see is a model in our brain based on many failable inputs. In that case, we simply can't simulate reality to any high degree anyway, or brain simply isn't big enough to understand the 10^33 particle interactions around you. But much more importantly, we don't have to, all we need it something to stop us walking off a cliff, and that cliff is there in most of the "worlds" so if we just see it as "one thing" then evolution has done its job.
There have been any number of attempts to define this more rigorously by people with significant brainpower, but none of these appears to have gained widespread acceptance any more than explanations for wavefunction collapse.