1) what is Trevorcard's relationship to the brotherhood? In the LoS2 DLC, the brotherhood knights were enemies.
Probably enemies, although He would likely help them against Dracula anyway. Who knows. most of Revelations is Dracula's cut content, so it's hard to say if he would actually have fought the brotherhood otherwise. The memorial at the hotel/residential building seems to imply that despite "Alucard the Vampire" being a VERY successful movie, the Church disliked it, for obvious reasons. So I assume The Brotherhood wouldn't be entirely ok with him. That said, if he's on their side and they know it, who can say? After Gabe fell, it seems they became Zealots, as evidenced by the heavy soldiers being if I remember, former criminals or something, and as such are vicious and brutal
2) What would Trevorcard use as a weapon? His sword, the crissagrim, is what is keeping Gabula in a dormant state. So, Trevorcard can't use it as a weapon.
if it's between moF and LoS2 then he can use it. Where do you think that 500 year gap comes from? LoS2's prologue takes place 500 years after MoF. As long as the game takes place before the prologue, he can use the sword.
3) Why would Zobek help him? Better yet, why would accept Zobek's help?
He wouldn't. he'd likely be in hiding, occasionally sending Necromancers after him/Belmonts, for their combat crosses or something, like in MoF.
4) Who would be the antagonist?
Dracula. As long as it's before the Prologue of LoS2.