Can I put Inkay into play, confuse it somehow, and use its "Upside-Down Evolution" Ability all on the same turn?
Yes, you may.
Upside-Down Evolution



x2
-20Upside-Down Evolution
Once during your turn (before your attack), if this Pokémon is Confused, you may search your deck for a card that evolves from this Pokémon and put it onto this Pokémon. (This counts as evolving this Pokémon.) Shuffle your deck afterward.
Confusion WaveBoth Active Pokémon are now Confused.
Can I put Inkay into play, confuse it somehow, and use its "Upside-Down Evolution" Ability all on the same turn?
Yes, you may.
If I use Inkay's "Upside-Down Evolution" Ability, does it remain Confused after evolving?
No, it does not. The phrase "this counts as evolving this Pokemon" refers to evolution rules that remove Special Conditions on that Pokemon.