If one of my {Dressrosa} type Characters has an [On K.O.] effect and is K.O.'d during my opponent's turn, does this card's [Opponent's Turn] effect activate first, or does that Character's [On K.O.] effect activate first?
In this case, you can activate the card's [On K.O.] effect and this Leader's [Opponent's Turn] effect in any order you choose.
If one of my {Dressrosa} type Characters is K.O.'d by my opponent's attack, can I use this [Opponent's Turn] effect to draw 1 card?
Yes, you can.
During my opponent's turn, my "OP10-090 Franky" was K.O.'d, and I activated its [On K.O.] effect to play "OP04-092 Rebecca" before the activation of this Leader's [Opponent's Turn] effect. In this case, which effect activates first: this Leader's [Opponent's Turn] effect or the [On Play] effect of "OP04-092 Rebecca"?
In this case, this Leader's [Opponent's Turn] effect will always activate before the [On Play] effect of a Character played using the [On K.O.] effect of "OP10-090 Franky".
I have "OP04-089 Bartolomeo", a card with a base cost of 3, on my field, which currently has a cost of 4 due to this Leader's effect. During my opponent's turn, my opponent uses the [On Play] effect of "OP02-106 Tsuru" to give Bartolomeo −2 cost. What would Bartolomeo's cost become in this case?
In this case, Bartolomeo's cost would become 2. Usopp's effect is a permanent effect, meaning it is processed first: this raises Bartolomeo's cost from 3 to 4. Tsuru's [On Play] effect, which is an auto effect, is processed after that, dropping Bartolomeo's cost from 4 to 2.
I have "OP04-089 Bartolomeo", a card with a base cost of 3, on my field, which currently has a cost of 4 due to this Leader's effect. During my opponent's turn, my opponent uses the [On Play] effect of "OP07-080 Kaku" to give Bartolomeo −3 cost. What would Bartolomeo's cost become in this case?
In this case, Bartolomeo's cost would become 0. Usopp's effect is a permanent effect, meaning it is processed first: this raises Bartolomeo's cost from 3 to 4. Kaku's [On Play] effect, which is an auto effect, is processed after that, dropping Bartolomeo's cost from 4 to 1. As this means that Bartolomeo's cost is now 1 or less, Usopp's effect would no longer apply to it; it would lose the +1 cost from Usopp's effect, so its cost would become 0.
I have "OP04-089 Bartolomeo", a card with a base cost of 3, on my field, and it currently has a cost of 4 due to this Leader's effect. During my opponent's turn, my opponent gives a DON!! card to "OP03-078 Issho", then uses Issho's [Your Turn] effect to give Bartolomeo −3 cost. What would Bartolomeo's cost become in this case?
In this case, Bartolomeo's cost would become 0. As the effects of the turn player's cards' take priority, Issho's permanent effect will be processed first, dropping Bartolomeo's cost from 3 to 0. The effects of the non-turn player's cards will then be processed, but Usopp's permanent effect will not be applied to give Bartolomeo +1 cost, meaning that its cost will remain 0.
