Weavile

If I use Bramblin's "Ride the Wind" attack and flip heads, can I bring up Weavile and use its "Assaulting Hunt" Ability?

Yes, that'll work because the Assaulting Hunt Ability is still happening during your turn.

If I bring up Weavile with the "Assaulting Hunt" Ability after my opponent KO's my Active Pokemon, can I switch in 1 of their Benched Basic Pokemon to the Active Spot?

No, because Assaulting Hunt only works "during your turn"

Due to a localization error in the Brazilian Portuguese version of Weavile (SV:Paldea Evolved, 134/193), the text of the "Caçada Ofensiva" Ability will be updated. The original text of this Pokemon's Ability was, "...você poderá mandar 1 dos Pokémon no Banco do seu oponente..." ("...you may send 1 of your opponent's Benched Pokemon..."). The text must be updated to match the intended effect of this Ability, which must be in line with the original Japanese text. The new text for Weavile's Ability will be, "...você poderá mandar 1 dos Pokémon Básicos no Banco do seu oponente" ("...you may send 1 of your opponent's Benched Basic Pokemon..."). The change will take effect immediately, and all Brazilian Portuguese versions of Weavile (SV:Paldea Evolved, 134/193) will use the updated Ability text.