If my opponent has no Weakness and I use Meowstic's "Perplexing Eyes" attack, will I do extra damage on my next turn due to Weakness?
If a Pokemon has no Weakness printed on the card, Perplexing Eyes will give that Pokemon a x2 Weakness to Psychic.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02), TPCi Rules Team (2019-05-16)
When I use Meowstic's "Perplexing Eyes" attack does it change my opponent's active Pokemon's weakness before I do damage?
No. Do damage first, then apply effects.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)