Do weather-boosted spawns appear on the "nearby" radar?

Yes. All spawn with the relevant typing are "weather" boosted, and they show up in nearby. I also see dratini's in nearby sometimes. I think it is just random.

Any mon has a chance of not showing up in nearby, from what I've seen it looks to be pretty random. They could also be close to despawning, which always takes them off the radar 2 min before the despawn.

Most of my nearby is weather boosted mons at the moment.

I've had the same issue - coincidentally, I've had it with Ralts too. Mostly it's the Common ones - Weedles in particular - that don't seem to be showing up on the Nearby. I've had 5 Weedles on my screen and only 2 in the Nearby window a number of times (and I now have 4-5 Level 35 Beedrils to toss into gyms as throw-away defenders, yay!).

It might be a function of lag - the weather changes, it prompts new spawns, and there's a lag between the new spawns and the Nearby screen. For Ralts, we'd never notice it because RALTS CATCH IT NOW CATCH IT NOW CATCH IT NOW!, but for Weedles and others, it would be interesting to sit back and just let things play out a few times to see how long it takes to make it to the Nearby screen. If it does at all.

But I ain't gonna do that with a Ralts! No way. MAYBE with an Eevee... after I've tapped it to see if it's a low CP Eevee. Maybe. (got a few Level 35's of them now too. Bad IV's, but level 35!)