Nat Dex AG still follows the rule of in-game obtainable mechanic so you couldn't use anything that's not legal in-game. You can try out Custom Game (challenge-only), Gen 7 Balanced Hackmons or Gen 6 Pure Hackmons which don't have this rule so you can start with any Castform weather forme or hacked stuff.
As for the weather, it's just like what TastyMushroom said, you have to use the respective weather-causing moves or a Pokemon with Drought / Drizzle / Snow Warning ability respectively. .