There is no cooldown on summoning your Guild Page/Herald, however the Guild Page/Herald, along with the Argent Gruntling/Squire have an internal CD to use their special ability (IE: Mailbox, Bank, or Guild Bank, etc).
Now, in terms of stoning you should be able to stone everything except the Sky Lantern (it's bugged), however, there really is no point, only the Argent Squire/Gruntling will show up as rare on Warcraftpets, and you don't need to have them stoned in-game to have them show up as rare on here. The other pets may show up as rare in your pet book, but not on the Armory or Warcraftpets.
Now, some people have OCD and need all the pets in their book to show up as rare, regardless of whether or not they show up as rare on the Armory or Warcraftpets. That's cool.