Well, the worst case scenario of Stonewall's death at BR 1.0, as accounted in What If?, is a possible collapse of the CS army comparable to the collapse of the US army OTL.
Whether (as the WI essay predicts) the AoP could be reorganized in time to take advantage and lead to the "quick war" all expected is questionable, IMO. As my old CW prof Dr. Robertson put it, BR 1.0 was a "clash of armed mobs" rather than a true battle of armies.
Rather I'd assume it takes a while to reorganize the army, particularly with a throng of noncom gawkers clogging the roads trying to congratulate the "victors". After let's assume weeks the AoP moves south and we have a second battle somewhere deeper in VA. Let's assume (following the RR in route around the rivers to hit Richmond) Culpepper, Orange, or possibly as deep as Gordonsville. Continuing Union victories could lead to an early siege of Richmond, otherwise CS victory drives the AoP back to Manassas or possibly all the way back to MD if severe enough.
Bee's survival, can't say. Considering the deep doo-doo his corps was in at the time of the (possibly ironic) "Stonewall" comment I can't imagine he'd be too pivotal, so you'd need to extrapolate his performance into future battles and there's little evidence to extrapolate from.