Yeah, hate to say but most of this thread is being done really wrong. Teddy Roosevelt and Andrew Jackson would be obvious favorites if this were a 1v1 tournament, sure, but it isn't. It's all of 'em in the ring simultaneously. And I'd say it's pretty hard to make a concrete argument for any of them - for the most part they're indistinguishably good at negotiation, which would matter far more than their actual physical capabilities.
I think it's hard to say. I'm having trouble logically justifying one "right" answer here. The first step is obvious - a 1v1 disadvantages the majority of the presidents, so those presidents would team up. Since each individual member, though, would not want to be the one stuck having to engage the strongest presidents directly, the resulting teams would operate defensively, as no member has incentive to try to engage the other presidents directly (ergo no incentive to launch an offensive). The strongest presidents also would not go on the offensive, because it's likely that they would, at best, trade life-for-life - not helpful when you only have one life. So the different factions would remain on the defensive. The problem is the next step. I can't see the heavyweights teaming up in response, but they certainly wouldn't take the chance to try to take one another out, since any in-fighting would probably lead to the death of both of them.
Much like Diplomacy, it should be an X-way draw, where X is, of course, the number of resultant factions of presidents. The trick would be which presidents could successfully fool their arenamates into ignoring the logic presented above and get them to act against their self-interest. Since such action would invariably lead to an offensive, there can be only two outcomes - someone dies, advancing the fight to an endgame, or the offensive stalls with no casualties on either side, in which case the situation reverts to the stalemate before.
tl;dr: Should be a multi-faction stalemate; whether it is or not depends on whether the presidents can successfully talk one another into ignoring what "should" be.