Sorry I wasn't clear before. For pets like the hatchling you have access to it on all toons once you have it on one. For example the Dark Phoenix Hatchling works the same as the Thundering and once you have it in your pet journal you can pull it out as a companion and battle with it on any toon on your account even if you later delete the toon that bought it.
As far as the trading goes, yes they have to be cageable. One party will roll a new toon on the other persons server and cage the pets from their pet journal to trade to the other persons toon.
Again, apologies if my previous post caused any confusion.