Was it possible for General Pope and the Army of Virginia to beat General Lee and the Army of Northern Virginia?
Possible yes. Likely, no.
If he did, what would be next, would he push onto Richmond?
He might try. I doubt if such an advance could succeed.
Immediate consequence - Lincoln issues the Emancipation Proclamation.
What about the Army of the Potomac, would they get a more aggressive commander and finally get rid of Mcallean?
Pope is confirmed in his command of the Army of Virginia. McClellan is downgraded in comparison. Possibly he is subordinated to Pope, or transferred to some other assignment, with the rest of his troops given to Pope. (He might resign in a tantrum if subordinated.)
Would they kill or capture Lee or any of his other generals?
Lee? Very unlikely. Some subordinate general? Possibly. There were a few dozen generals in the Army of Northern Virginia, and many of them were KIA/CIA, even in Confederate victories. For instance, BG James Archer, captured on the first day of Gettysburg, despite the rout of most Union forces that day. During OTL Second Manassas. MG Richard Ewell was badly wounded and had a leg amputated; MG William B. Taliaferro was also injured (two of Jackson's three division commanders).
It depends on how much of a victory the battle is. Let's say that Pope leaves enough troops on his left to hold off Longstreet, and that he somehow gets all his attacks on Jackson to go in together. Kearny's division turns the flank of AP Hill's Division on Jackson's left, causing a stampede. Jackson and Hill are wounded while trying to stem the rout; Hill is captured. Lawton, Jackson's senior surviving subordinate, takes command of Jackson's corps and orders a withdrawal to the south.
Meanwhile Longstreet's attack on Porter's V Corps stalls with minor success. As night falls the Union holds most of the field, and Lee retreats to Warrenton Junction. Union casualties are 1,500 KIA, 7,500 wounded, 1,000 MIA. Confederate casualties are 1,500 dead, 7,000 wounded, 3,000 MIA.
Would the war end early? Let me know what you think.
Pope would basically replace McClellan in the East. McClellan's failure in the Peninsula would be contrasted with Pope's victory. However, Pope was not likely to be a long-term success. He was an outsider who had alienated his new subordinates and men with dismissive comments. Making good those comments with a victory would help... but he had the wrong personality - a thoughtful commander wouldn't have talked that way in the first place.
His orders and decisions at Second Manassas were not well-made. If he had won, it would be more luck than skill. He'd think he was clever, though, which would just encourage his arrogance.
IMO, when he tried to march on Richmond. he'd blunder into an operational ambush and get badly beaten. Then he gets replaced by someone more competent; possibly Porter, except that Pope is likely to get rid of him. Sumner? Franklin?
Not Banks or Sigel.
Effects on the war: the EP brought forward by a month. McClellan gone. Lee's victory streak broken.
Yes, I think the war ends sooner. Lee will tend to stay on the defensive in Virginia; but he may lose another battle. OTL, Lee didn't truly lose a battle on the strategic defense until the storming of Petersburg. I don't see that changing. There will be attrition, though. Secondary battles will go both ways, Lee will have some counter-attacks fail.
If Pope's replacement is able, then the Virginia campaign may end in late 1864 as the CSA is exhausted.