I think it sounds like a very interesting idea. I'm all for it so long as it's implemented well!
Like others have mentioned, players should be able to change the weather when using a pet that alters it (reverting back once the player's weather ends). PVE only.
It would definitely add another layer of complexity to Tamer battles, and sounds neat in theory but may not work out as well in practice. Some things to consider: Would it replace any weather that the Tamer pets already use? Will it feel like it "forces" the majority of players to use a weather changing pet when completing Tamer dailies? Or will there still be an element of choice, where non-weather changer teams are still viable?
If there are other issues that can't seem to be resolved well enough to keep things fun and balanced, perhaps the environmental weather should be applied to wild battles only.
For Cleansing Rain, how about Moonglade? It's an overcast Druid-y area (healing-centric), so maybe it would fit?
Call Lightning (Lightning storm) could be applied to Netherstorm (along with Arcane Winds)? Definitely Isle of Thunder though.
I think Acidic Rain would need its own zone, even though it eventually causes Cleansing Rain.
What about a few weather effects being applied to a single zone (if appropriate)? Might add some diversity for wild battles, and you never know for sure which weather you'll get.