BlazBlue:
R1: If Cole starts off alongside Ragna when they arrive at Kagusutchi (Start of Calamity Trigger) then I am pretty sure Cole would help Ragna as much as possible. Though with Cole's feats wouldn't get far due to the hax in this verse although he would be a good companion for Ragna to burn out his angry side and maybe (Since Cole is in his good karma) can help Ragna to calm down and will assist in killing Terumi and stoping Izanami's Doomsday plans.
R2: Assuming Cole drops in before Izanami's plan in Chronophantasma. Cole tries to live a normal life at first, probably going back to the bike courier skills, his powers and bravery helping him deliver packages. Though this might attract the N.O.L about his powers due to not them coming from a Nox Nyctores, nor an Ars Magus or a Grimoire, making Relius curious if Cole came from the Boundary and would have people to try and capture him. This would leave him wanted by the world like Ragna and soon would get involved in stopping the N.O.L.
R3: Evil Cole runs around wreaking havoc. N.O.L soldiers, mercenary, and monsters, Cole strikes them all down. He ends up living in one of the distant Heirachical Cities, preferring to live in the monster ghost town, as the place reminds him of Empire City and New Marais, a place of no rules, where the strong are free to do as they please. He, would possibly get temped to join with Terumi and Relius in their plans to get more power and he would go stomping heads with Ragna where he would feel a connection and would act like Terumi to get a rise out of him.
R4: Beast Cole kinda goes on a rampage and starts destroying everything leaving nothing but destruction like the Black Beast many years ago. Hakumen, Jubei, Good Cole and Ragna coming to stop Beast Cole but Ragna would probably lose control and turn into the Black Beast leading well...end of the world. Though with Good Cole around this might change with Good Cole helping Ragna to destroy Beast Cole and making sure Ragna never loses control while Terumi and Izanami watch from the safe lines. Then everything continues like in R1.
(Sorry if I necro this...I was hoping to join in the fun)