Does our Sun have a counterpart, i.e., is it a part of a binary-system? If so, how does the other star look like and where is it?
There is no observational evidence that the sun is a member of a binary (trinary, or more) star system, where "star" means an object that is at least ~80 times the mass of jupiter and emits energy/light via standard hydrogen fusion.
Some evidence that people point to is that the majority of stars in the Galaxy (perhaps 60% or so) are binary. However, that does not mean that we are a binary system, just that we are among those which are not.
There is the Nemesis star idea which was a hypothetical binary companion to the sun to account for periodic mass extinctions roughy every 26 million years (as in, the two stars would therefore be on a 26 million year period, so the companion is a few light-years away). There are two problems with that: First, that a 2010 reanalysis of the data showed that it was too periodic over the past 500 million years, during which time we've gone around the galaxy twice, and so our hypothetical binary orbits should have been perturbed and not be perfect (though others say that the precision of the geologic timescale isn't good enough to say this). But, second, we can see stars out to a few light years -- we can see small, faint stars out to thousands of light years. We have all-sky infrared surveys that should be able to pick up the faintest even star-like objects to at least a few light-years, and yet ... nothing.
The other problem is what LDC3 pointed out in their answer: We should see some systematic motion of our own star in orbit around the common center of mass of the hypothetical binary. This would not be a yearly wobble, but rather it would be a slow motion of the entire sky on top of the other >1 year motions that we see. We now have very accurate astronomical records dating back at least a century of star positions, especially close stars. Even if we were on a 26 million year orbit - and especially one much shorter as some people claim - we should see effectively that our star is making a small arc, a part of a circle as it orbits the center of mass of it and the binary. We don't. There does not seem to be any systematic signal in the motions of stars that require the binary star model.
So, to summarize: A binary companion simply lacks any observational data; if it existed, we should not only have been able to see it by now, but we should also observe not only the stars in our sky show a systematic motion due to our orbit around it, but we should again see THAT binary star also move very quickly, relative to other stars, as it orbits the common center of mass.