Is he able to wind down the war early?
I think without the McCarthy challenge of OTL that is unlikely. The Paris talks may not even begin before the election, and if they do they won't produce an agreement in time.
Is Nixon still his aponent?
Does who he defeat have any effect on his 3rd term?
On the other hand, didn't LBJ go in to self-distruction moad as soon as he left the WH, which probably didn't help him prolong his life. If he's still president he probably doesn't do that, so is it really a sure thing that he dies during his second term?
His health did figure into his OTL decision not to run according to Wikipedia.
If RFK doesn't primary Johnson, ther'll be no need to ask him to be Veep.
There's party unity, which would be fragile even without a McCarthy challenge. He can't pick McCarthy or Wallace or anyone from their factions because he disagrees with them too much, but if my understanding is correct and Bobby was somewhere between LBJ and McCarthy then he would be the best pick (not necessarily a good one, just the least bad) for keeping the Democratic Party together. And having him as VP might give him some peace from that direction during his 3rd term. Against this is obviously LBJ's antipathy towards the Kennedys (though, again according to Wikipedia, by this point he didn't like Humphrey either) and the hell that HH can raise if he's uncooperative.
Would RFK even accept a VP nomination from Johnson?
It would make him the confirmed heir apparent, which is a big plus. (Johnson was born in 1908, so I don't think Kennedy would expect his death.) On the downside, it would tie his future to what is currently a very troubled presidency. It's a great offer if Johnson doesn't screw up his 3rd term and a terrible one if he does, so I guess it depends on whether or not Bobby took all-or-nothing risks. The best thing to do IMO would be to put forward terms for agreement, but I'm not sure what they'd be.
Could there be an outside chance that an Alie of RFK's (or someone ideologically closer to Kenedy) ends up as Johnson's running mate instead, to unify the party?
Obviously, but I don't know who his allies where in '68.