Yep. Well, it's kinda like a half-joke here, I guess, and you've already correctly guessed a lot stuff.
Space is mostly empty, everybody knows. It mostly is. The solar wind stream is like 99%+ unbound protons and electrons (I could go look this up, but my new anxiety therapy is actually quarantining myself to only certain realms of the internet, too, as is self-deprecation on a species-wide scale) blasted away from the sun, sure, but it's way under a trillion trillion times less dense than the air we're breathing right now. You nailed it: we wouldn't feel it at all, but of course it's quite undesirable for astronaut wear-and-tear. But it's not the usual solar wind particle and photon energy spectrum that will kill you quickly from radiation poisoning; There can be solar events that generate deadly radiation, usually at least once in every five years-ish, but highly variable. Mother Earth shields us here on her surface from it all :).
And no one really predicted the solar wind, I think. I remember how it falls out of the maths as one of four possible solutions for a family of magnetohydrodynamic equations. We're magnetically shielded basically up to 10 earth radii outward, so we had to send something out far enough to see it. Well, or you could have an orbit over the poles. You're kinda out in the solar wind over the poles. Sputnik's orbital inclination may not have been high enough to see it. But, naturally, Sputnik wasn't outfitted with instruments to measure the solar wind anyway, though you may still see secondary effects, if you know what you're looking at (we didn't). Plasma does all sorts of terrible things to spacecrafts, and it's literally everywhere in space. Even so, the separate garden variety space plasmas of different regions are quite distinct in their characteristics and behavior, we know now.
Lunar dust is much heavier than solar wind, larger particles, and there are plenty of studies on what the solar wind does to exposed regolith.
Mars used to have an atmosphere. When it lost its magnetic field (which, let's just call that process "magic", for now), solar wind came in and stripped the air away. And it still would today, if we tried to artificially enhance Mars's atmosphere.
Probably all "main-sequence" stars, like ours is, have their own solar winds. That's why my friend is looking for exoplanets with a very particular type of radiation produced by the interactions between a planet's magnetic field and the solar wind. We should probably only colonize an exoplanet with a sufficiently strong, dipole-shaped magnetic field oriented in a certain direction. That's on top of all our other needs. I mean, really, what a pathetic species we are. Shelter in place, kids.
Please ask questions or leave comments, I’m trying to get better at this :).