I doubt Blackbeard would execute Garp, if only because there's no profit in that for him. I mean, he'd actually be better off "selling" Garp to Cross Guild in order to collect his bounty, rather than killing him off.
And so far, his entire stated plan has been to try to ransom a high-profile Marine back to the World Government, in order to get them to recognize Hachinosu as a member nation. That's what he wanted to do with Koby, and that's most likely what he's planning to do with Garp. Though, I wouldn't be surprised if Blackbeard's planning a few little twists on that, that we aren't aware of yet. But, killing Garp would actually be detrimental to his stated plan.
What would Blackbeard gain from killing Garp?
I guess it would take him off the board, which could benefit his long-term plans. It'd be one less major fighter he'd have to contend with later on down the line. The Marines losing Garp would significantly weaken them, and that's not nothing.
Maybe Blackbeard could force feed Garp two "junker" Devil Fruits and turn him into a living bomb to destroy New Marineford or Mariejois from within? But, that would mean sacrificing two Devil Fruits, and even if their powers weren't anything Blackbeard would want, I'm not sure if Blackbeard woudl do that. Devil Fruits are still pretty rare, and I'm not sure if even Blackbeard has two spare ones just lying around. Even if they were the..."Bow Tie" Fruit and the "Dung Beetle" Fruit or something. Plus, that wouldn't really be a "execution", it'd be more of a Trojan Horse situation.
I suppose a public execution would make for a pretty decent distraction. Having this big, public display that focuses the entire world's attention on one spot, while Blackbeard actually makes a key move in a completely different place entirely. That does fit Blackbeard's MO. It could make a good distraction, while Blackbeard makes a move on Mariejois or something.
But, I think there's a more interesting possibility.
Blackbeard has Laffitte on his crew. A guy who can hypnotize people. And the Monkey Family seem VERY susceptible to hypnosis. Since they're kind of dumb. Right now, Laffitte seems to be keeping a eye on the Mariejois situation for Blackbeard, but he could join back up with the main crew pretty soon.
Could Blackbeard have Laffitte hypnotize Garp into joining his crew?
I don't know. Maybe Garp has too much "willpower" for that sort of thing to work on him. But...I could see it happening. Even if he's basically "sleepwalking" through battles. Garp is strong. But, he's also kind of dumb like Luffy. I could see Garp being susceptible to hypnosis.
This way, Blackbeard could still ransom Garp back to the Marines. But, then he'd have a secret agent within the World Government that he could use down the line, too. A sleeper agent that he could activate at his whim.
It feels like something Blackbeard would do. I could see it happening. I'm just not sure if it's actually possible for not. The x-factor here is if Garp's Haki can shrug off the effects of hypnosis.
As for who would try to rescue him?
Well, I mean, Koby is the obvious choice. He's a total goodie-goodie. Garp's basically his mentor. There's no way Koby wouldn't at least TRY to rescue Garp from Blackbeard. And his friends would all probably follow his lead.
Maybe Sabo and/or Dragon? This would be pretty unlikely at the moment, due to the Revolution being preoccupied with the Siege on Mariejois. But, Dragon may want to save his dad. And Sabo likes to make a show of things. I could see Sabo wanting to make a move against Blackbeard, if only to avenge Ace. Saving Garp would just be the cherry on top of Sabo's revenge sundae.
Luffy? ...No. No, I don't see Luffy trying to save Garp. If anything, I feel like Luffy would just feel bad for those poor Blackbeard Pirates who are about to get smashed into the ground. To Luffy, I think Garp is too tough for him to even think he's actually captured. I think Luffy would assume that Garp is just stringing them along until he can strike at them. Cruel jerk that he is, Garp can handle himself. Luffy isn't going to take time out of his journey to save Garp. Unless the Strawhats have to go to Hachinosu anyway to get the last Rio Poneglyph, or to save Pudding. If Luffy's there anyway, MAYBE he'd save Garp...reluctantly. But, Luffy isn't going out of his way for it.
The Whitebeard Pirate remnants? If only to stick it to Blackbeard? Eh...meh. I don't think so. The former Whitebeard Pirate members are too scattered and too broken by their defeats to reform at the moment. I don't see it happening.
Sengoku and Tsuru aren't impossible. They are Garp's old friends. But, if anything, I could see them joining up with Koby on his rescue mission with SWORD, rather than going on their own.
Smoker? Nah. He might not be strong enough for one. Plus, he's got his own, ill-defined STUFF going on.
Would Akainu try and save Garp? I think he probably would. This whole situation is a embarrassment for the Marines, and I could see Akainu deciding to settle this personally. And Akainu probably HATES Blackbeard, since he hates all pirates. If he could get away from his paperwork, that is. And that's really the biggest issue. With the Siege of Mariejois, Akainu just might have his hands too full to save Garp himself.
What about Kong? We haven't seen Kong do...ANYTHING...EVER! So, this would be a good opportunity to see why he's the big boss of the Marines. We could finally see his Devil Fruit, if he has one. But, he shares a lot of the same problems as Akainu. He might be too busy with Mariejois, at the moment.
CP0? I mean, secret rescue and assassination operations are kind of what they're built for. I could see it. But, I doubt it, since we've kind of already explored CP0 already via Egghead. CP0 is pretty well traveled ground at this point. I think Oda would want to give the spotlight to someone else.
The God's Knights? Well...maybe. We likely haven't seen all of them yet. Rescuing Garp would give Oda his excuse to design another dozen or so new characters, which he would love. It's possible. But, again, they're a little distracted with Mariejois. I feel like we're more likely to see more of them fighting the Revolutionaries, rather than rescuing Garp.
Shanks? I mean...hmm. This would be interesting. I suppose Shanks and Garp HAVE met before, at least at Foosha Village. I...don't know what their relationship would be like. And Shanks probably is looking for his chance to attack Blackbeard. Shanks...would be a VERY interesting choice. And it'd give us the Shanks v Blackbeard fight we've all been halfway expecting. Just in a way we probably never expected to see it. Which Oda would probably love. Maybe even getting to see Garp and Shanks team up against Blackbeard, like Roger and Garp teamed up against Rocks on God Valley? I'd like to see this. But, I don't think it's terribly likely.
So, my best guess for a rescue party would be Koby and his friends. Maybe with Sengoku and Tsuru tagging along as backup. It's not the most interesting option, but it would be the most likely option. Mind, I doubt Garp is getting rescued, anyway. It's probably more interesting to let Blackbeard get farther along in his master plan.
