Will the move cure my Pokémon’s status condition even if the other Pokémon doesn’t receive it (if it already has a status condition or the move doesn’t affect it)? Does it have to actually give the other Pokémon the status effect to cure mine?
It will not transfer volatile status conditions, such as confusion. It will fail if the target is immune to the status condition (such as via type, Ability, Safeguard, or by already having a status condition).
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I also tested it out on PS! For Burn and Toxic.
So that confirms the answer for you.