Cell said it was sheer, unexpected luck that the core remained intact though. Seems like it was a one in a hundred chance that core was unruptured by the explosion.
That'll be a really really flimsy explanation for why he didn't use it against Beerus considering he'd lost all self control during his rage boost, so him exercising enough restraint to avoid using SSJ3 would be horrid writing.
I'm not that sure. Their rivalry doesn't seem that serious, but then again, Goku doesn't take his rivalry with Vegeta all that seriously either so maybe they would.
I go with a secondary core existing inside his lower body to avoid an obvious weakness of his head, and a connection between the two keeping the secondary core inert when his body is more intact.
Let's say Gohan didn't throw himself in the way of SPC's blast, causing Vegeta to be killed. How does this affect the events of the Boo arc and onward?