I have heard it claimed the Bull run/Manasas could have gone the other way.
I also suspect that with different luck and strategy Richmond might have fallen early in 1862.
This seems to be two questions. Is the second supposed to be an implication fo the first?
My $0.02 is that if First Manassas had been a Union victory (which was entirely possible IMO), it has consequences all across the country, not just the Virginia theater. So the early fall of Richmond would not be the main effect.
In July 1861, the CSA was still very much of a (damn, can't think of the right word).... start-up, prototype, outline, trial?
Few Southerners had made strong physical or emotional commitments to the new government, which was still building out its organization.
There was a wave of enthusiasm, to be sure, but it was in large part driven by the braggadocio and gasconade of the Fire-Eaters, who talked wildly of how "gallant Southern gentlemen" could whip any number of "pasty-faced Yankees". OTL Manassas, as it happened, and especially as it was reported, seemed to confirm all this talk. It gave the CSA a huge psychological boost.
It also put a check on Union operations, and gave the CSA about six months to continue its organization. During that period, Southerners "worked in" to supporting and obeying the CSA. When the Union offensives of early 1862 took New Orleans, Nashville, and Memphis, and threatened Richmond, the CSA had the resilience to survive.
If Manassas has the opposite outcome, with "Southern gentlemen" fleeing in rout, or taken prisoner and paraded before jeering crowds in Washington... the entire secessionist enterprise could unravel right then. Almost certainly, Kentucky would declare for the Union. A lot more men would adhere to the "Union government of Virginia" being organized in Wheeling. The Five Civilized Tribes in the Indian Territory won't ally with the CSA.Tennessee and North Carolina will have second thoughts.
As to when, or whether, Union troops capture Richmond... Assume that despite the Manassas disaster, the CSA carries on.
McDowell's Army of Virginia wasn't capable of a long overland campaign; it was poorly organized, mostly due to its newness. The CS army is worse off, but it's on the defensive, which is easier. Still, it's possible that the Union gets to Richmond before the end of the year. For one thing, the CSA will lose resources, due to disaffection, and send assets to the West and elsewhere much sooner.
I have no idea what route McDowell and Scott had considered for a march on Richmond, if they had even thought that far ahead. The CS army would retreat SW, along the railroad, but Richmond lies due south, via Fredericksburg. It may end with an ineffectual siege of Richmond, starting as early as November 1861. Or if the CSA is bleeding resources elsewhere, and McDowell can get army's act together, Richmond could fall by December.
1) Does this end the rebellion?
See above. If Richmond falls, the unraveling will start. Not long after, the Union will take New Orleans.
2) If the Confederacy is defeated before the Emancipation Proclamation how long does slavery survive
About 10 years, maybe 20 at the outside. "Conservative" emancipation, with blacks remaining non-citizens, is the most likely path.
3) What happens to government in the rebel states
The secessionists leave office, and local unionists form new governments which are recognized by Lincoln.