Lincoln wanted a quick reconciliation with easy terms for the South after the war, yet Radical Republicans wanted to punish the South, and eventually got their way after Lincoln’s assassination.
This is one of the better "what if's?" that could be applied to speculative history. Although Lincoln would have continued to have been hated in the South, there is a great chance that he would have been re-elected several more times had he been willing to run (probably equalling or surpassing Franklin Delano Roosevelt's four terms). The United States would not have had to endure the term of Lincoln's weak successor, Andrew Johnson; additionally, the incredibly corrupt terms of U. S. Grant may have also been eliminated. Of course, if Lincoln's less punitive Reconstruction policy had allowed Southerners to regain their rights with more immediacy, the Southern vote might have defeated him. I like to think that the imposed social reforms of all Southerners--especially to the freed slaves--would have gone more smoothly with Lincoln as president for a longer period. Whether the Ku Klux Klan and Jim Crow laws would still have reached their terrible heights is, of course, a question we can only ponder.
For an excellent piece of speculative/alternative fiction that addresses this issue, you may want to read Harry Turtledove's novel, Guns of the South. In his revisionist version of the future, the Confederacy lives on, defeating Federal forces; Lincoln loses his re-election bid; Robert E. Lee is voted president of the Confederate States; and Nathan Bedford Forrest (the first leader of the KKK) takes over overall command of the Southern armies.
You have identified the major likely difference if Lincoln had lived. It is likely that Reconstruction would have gone much differently if he had not been assassinated. However, it is hard to know for sure how things would have changed.
As you say, Lincoln wanted to be much more lenient to the South than the Radical Republicans did. If he had lived, it is quite possible that Radical Reconstruction would have happened. The Radical Republicans were able to run over Pres. Johnson, but it would have been much harder to do that to someone who was as strong of a leader and as popular as Lincoln. That means that Lincoln would probably have gotten his way and his more lenient version of Reconstruction would have been implemented.
But what difference would that have made? After all, African Americans did not really end up benefitting much from Reconstruction. What would a more lenient Reconstruction have done? Maybe it would have allowed the South to be harsher towards the freed slaves, though that is hard to imagine. Maybe it would have made the South less angry at the North and so there would have been less of a solidly Democratic South that hated the North.
As with any counterfactual, it is impossible to know for sure.
I would expect that this is a question of perception. Had Lincoln not been assassinated by John Wilkes Booth, the South may not have been driven to its knees in such a demoralizing way. I don't believe that Lincoln would have wanted this to happen: in my mind he wanted the division between the States to be healed, offering as much help and dignity as possible. I wonder if the hatred of many people in the South for blacks would have been so great if they had not felt their livelihood was taken from them in such a radical fashion. That the slaves needed to be emancipated, there is no question. However, had the South been given some federal assistance to still farm their lands—hiring men and women of color to work the fields rather than enslaving them—perhaps the transition of the U.S. to a nation where all men and women were free, regardless of race or color, could have taken place in a less "painful" way.
Lincoln's pragmatism and his general approval from the public was a very powerful force in his administration. Considering the ill-will caused by assassination of any stripe, it would be hard to argue that anything but good would have come from his living.
Think about the efforts he put forth to really change the way Americans viewed each other.
At the time, there were no term limits for the Presidency. Although Washington's example continued for many to follow, it was possible to continue running for office indefinitely. Lincoln could easily have won a third term and continued his groundbreaking policies instead of Andrew Johnson's widely disliked term (and his infamous impeachment).
Reconstruction might have had a chance with Lincoln at the helm, as he could have at last tried to shepherd a radical Republican Congress towards something like realistic reform and reconciliation policies. He would have had a "honeymoon" period after winning the war in which he may have been able to achieve more than Reconstruction did under Johnson. We might also have seen some investment in the South's physical reconstruction vs. the almost immediate detour west. All of this being said, we might be giving Lincoln too much credit, and we might be underestimating the resistance in southern society towards progressive legislation by Lincoln or anyone else.
In response to #10, I doubt Lincoln would have been elected for a 3rd term; as I recall, he won the Election of 1864 only because of the fall of Atlanta, in which Northern voters approved of him as Commander in Chief. By the Election of 1868, I'm sure those in the Southern states would have panned him, and he wouldn't have been in the running for the Presidency. As others have noted, Southern integration back into the Union would have occurred more quickly had he been spared assassination; the political problems might have healed quickly, but the social and cultural ones would have lingered much longer, regardless of who was in the White House.
I like to think that because of Lincoln's plan to reconciliate the North and the South quickly and as amicably as possible, there would not be the bone-deep division of Yanks and Rebs that is still so prevalent today. I'll never forget as a child, the time my parents took my brothers and me to a Civil War reenactment in Tennessee. My brothers and I wanted to get hats from both the Union and the Confederacy so we could play "Civil War" at home, but my father said firmly, "You will all get hats from the Confederacy, or you will get no hats at all." This, of course, brought a wide grin and a nod of approval from the gift table owner.
Oh, I definitely think history would have been different had Lincoln not been assassinated! For one thing, he would have seen that The Reconstruction got carried out like he wanted it to, and the freed slaves would have been treated right. I really don't like studying about or teaching about The Reconstruction because of Andrew Johnson, the carpetbaggers, and all the unfair laws against the freed slaves! It's a really depressing era in our country's history, second only to the Civil War itself. Lincoln could have made a difference had he been allowed to!
I would imagine we would have had a much more influential and effective shepherd of Reconstruction policies had Lincoln served out his second term and Johnson had never been President. Congress would have been more willing to pursue an active Reconstruction policy past the Civil War Amendments and perhaps land reform or more secure political rights could have been achieved.
I also think that Lincoln would hold a much less revered place in our history in terms of his Presidency. As with Kennedy, his flaws are often overlooked given his assassination.
Speculative history is great, and one of my favourite examples is The Plot Against America by Philip Roth. However, to answer your question, Lincoln's survival would have meant he could carry through his agenda of political and social reform for longer, hopefully making it a smoother transition as identified in #3. But at the same time, his death made him a powerful symbol that must not be ignored in terms of its importance.
Lincoln is often remembered for winning the civil war and the Emancipation Proclamation. It is easy to forget that the country was not rebuilt in a day, nor were the former slaves granted equal status in the union. Lincoln's successor, Andrew Johnson, is often seen as a President that worked to undo or undermine the accomplishments that we credit Lincoln with achieving.
I must agree that I don't think a second term for Pres. Lincoln would not have changed the immediate post-Civil War conditions or attitudes very much. There was so much resentment about what the "other side" had caused on both sides that forgiveness and leniency never really had much of a chance of becoming reality. Human nature demands revenge...
I think had Lincoln been allowed to oversee the Reconstruction process we would have seen a smoother transition from slavery to non slavery as a way of life in the South. As someone else has mentioned it is likely there still would have been problems, but Lincoln was strong enough to stand up for what was right.