Or are there virtual W and/or Z bosons that make particles attract (or repel) each other without changing the interacting particles like virtual photons do, but the interaction is so weak we just can’t tell?
Anything the electromagnetic force can do, the weak force can do also. The problem is that, since the electromagnetic force is so much stronger than the weak force, at any useful distance, in any interaction where the electromagnetic force can work, its effects overwhelm the effects from the weak force so much that the weak force is unnoticeable. So for the most part, the only interactions where the weak force is noticeable are those where the electromagnetic force doesn’t work, and those are mostly decays or captures.
The weak interaction does attract and repel according to weak isospin, and that means that it also affects particles, like neutrinos, that don’t participate in the electromagnetic force. That effect is negligable, though. Unlike the electromagnetic and strong forces, which are mediated by massless photons and gluons, respectively, the weak force is mediated by massive W and Z bosons. In fact, they’re actually very heavy, with about two orders of magnitude more mass than the neutron. As a result, the weak force is short-range and, well, weak. It doesn’t form bound states, and I’m not aware of any processes (including ones where the electromagnetic force doesn’t apply) in which the weak force matters aside from decays and flavor changes. If you’re looking for some sort of kinematic effect, it’s going to be completely swamped out by other factors and totally immaterial.
Yes. In the case of Z boson exchange, neutrinos interact in just the way you are thinking. For instance, a neutrino can interact with an electron, trading momentum but leaving both particles intact. On earthly scales, you need precision detectors to observe this, but it’s readily done. On astrophysical scales it shows up in, for instance, supernova explosions, where the neutrinos produced in the core have macroscopic effects on the outer material as they pass through it.