If I have 1 of this Character on my field as well as 1 "EB04-038 Rosinante & Law", which has the effect "under the rules of this game, also treat this card's name as [Trafalgar Law] and [Donquixote Rosinante]", does that mean that since my Characters have a total of 4 different card names, this Character gains +4000 power from this card's [Your Turn] effect?
No, it does not. As there are two cards in play with different card names, it will gain +2000 power.
If I have 1 of this Character on my field as well as 2 "EB04-038 Rosinante & Law", which have the effect "under the rules of this game, also treat this card's name as [Trafalgar Law] and [Donquixote Rosinante]", for a total of 3 Characters, does that mean that this Character gains +3000 power from this [Your Turn] effect?
No, it does not. As there are 2 cards in play with different card names, it will gain +2000 power. Even if you have 2 "EB04-038 Rosinante & Law", they are not considered to have different card names.
If I have 1 of this Character on my field as well as 1 "OP13-031 Trafalgar Law] and 1 "EB04-038 Rosinante & Law", the latter of which has the effect "under the rules of this game, also treat this card's name as [Trafalgar Law] and [Donquixote Rosinante]", for a total of 3 Characters, does that mean that this Character gains +3000 power from this [Your Turn] effect?
Yes, it does. In this case, since there are 3 cards in play with different card names, it will gain +3000 power.