OK, my answer for the first part was either Shaymin or Celebi using Grass Knot (no other Pokemon could do the proper damage with the proper Speed without activating a message). I had forgotten to take into account Low Kick somehow (now I feel dumb!). So, bonus points go to those of you who figured out that Primeape would also work. I'm somewhat disappointed that no one considered expanding the pool of usable Pokemon by giving Pikachu a Macho Brace and Raikou a Choice Scarf, as that would completely remove the Speed restriction. I'm somewhat glad no one did, because I suspect that there would be more Pokemon that fit the solution.
Next time I guess I'll post every part of the message. Using Rest was just supposed to indicate that the Pokemon as at full health. No matter what you get a message when you use Rest, so I assumed that wouldn't be a big issue. I guess that's what I get for assuming (and writing the riddle when I'm pretty sleepy...). Expect another soon, and this time I guess I'll be a bit more thorough in my message writing. And this time, I'll consider each case a bit more carefully so I can guarantee that there is only one correct answer (or at least by 99.6% sure).
The "Rain continues to fall" message was put in there so you'd know that there wasn't sunlight (as Specs Sunny HP Fire could do a real number on Shaymin).