In combat mostly just the same old Wolverine except less rage and animalistic stuff.  Like the difference between someone trained in Krav Maga killing or incapacitating someone swiftly and ruthlessly and one of the typical Wolverine chop everyone up and scream a lot.  Less skewering someone and picking them up over his head, more cutting the legs from under someone and finishing them.

John Wick rather than Rambo.

In regard to leadership he's less the on the ground lead by example type he usually is.  I picture him as having earned the loyalty of his men in that way long long ago but after establishing himself in Madripoor he now had middlemen there that would be that sort of leader.  He's more the scary face and promise of retribution behind the organization than the actual boots on the ground sort.

For the previous questions,

Hex is basically using Hellboys big ass pistol yeah, The Good Samaritan.  If you meant the Red Right Hand I don't want him to have the actual kinda squared off giant hand I don't want that.  I thought about having him have a burned up right hand or maybe a red glove, but the scarred up face and resurrection are probably enough.

For Old Patch, yes basically and absolutely yes.  Going off the old like bushido shit Wolverine often had ties to there are crimes he will neither participate in nor allow anyone to participate in in Madripoor, but he's not a super hero anymore.  Maybe a hero to many in Madripoor but not actually a hero.  Sort of an anti-hero, means to an end for the law abiding in Madripoor.  Some love him and some accept him and are maybe even like grudgingly glad he's around in the same way folks might become accustomed to and fond of a dictator.