No I'm assuming that:
1. Lincoln isn't seriously injured (Booth also had a knife). Perhaps Lincoln doesn't dismiss the soldier guarding the booth box.
FTFY.
John Frederick Parker
WP was actually a police officer, and by all accounts one of worst on the DC police force.

However, even had Lincoln had several heavily armed, competent bodyguards, the chances that Booth, a famous actor, would have been refused entry would have been rather slight. In fact, Lincoln had previously requested to meet Booth after a particularly memorable performance (Booth declined).
After the Fact, by Fred Saberhagen, containes a plausible Lincoln survival scenario:
2. Mrs. Lincoln isn't seriously injured.
It's unlikely that she would have been, for a variety of reasons.
3. The assassination attempts on William Seward and Andrew Johnson were also unsuccessful.
Assuming Lincoln had survived his term, the consequences of Johnson's assassination would have been minimal, other than to increase the howling for revenge from many in the North.
Well I'm not one of those who believes Lincoln would pander to the demands of the Radical Republicans but he'd support a plan that was stronger than that Andrew Johnson advocated.
Actually, Lincoln favored a moderate reconstruction; see his Ten Percent Plan
WP.
I suspect he'd also support full citizenship rights (including full voting equality) for black Americans. He'd likely support the 14th and 15th Amendments to the US Constitution.
I imagine that he would have come around to these ideas eventually.
However he'd also favour a quick reintegration of the southern states into the Union, and full citizenship for all former Confederates except perhaps for a few of the most prominent or those who tried to use violence and intimidation against former slaves.
He also exempted from his
blanket pardon "all who have engaged in any way in treating colored persons, or white persons in charge of such, otherwise than lawfully as prisoners of war" (IOW, war criminals).
Now getting off the ACW let's look at economics, a rather more important matter. Lincoln himself greatly distrusted Big Business and, given the rapid industrialisation of America in the decades after the ACW, perhaps Lincoln would have attempted to institute policies to alleviate the negative impact on the working classes. He did at times made statements almost akin to socialism.... (Something Herr Turteltaub got right).
I think that Lincoln's socialist leanings have been overstated by some, including Turtledove.
Let not him who is houseless pull down the house of another, but let him work diligently and build one for himself, thus by example assuring that his own shall be safe from violence when built.
(
Source)
Of course there's also the question of how well a surviving Lincoln would be able to work with Congress; he might well have won the war but lost the peace.
I think he would have been able to work reasonably well with Congress; the Radicals were never an absolute majority.
Would he have broken the two term tradition (I rather think he would).
Can you elaborate on that? I've always assumed he wouldn't have.
And who'd have succeeded him?
Again, there's no good reason to believe it wouldn't have been Grant.
Presumably you're referring to Joshua Lawrence Chamberlain
WP. In 1868, Chamberlain's only political experience had been two years as governor of Maine, and, although a war hero, he wasn't a well-known national figure like Grant or Sherman.
His successor would have had to deal with the 'second generation' of reaction from the former Confederates; remember most of the 'Jim Crow' laws were enacted in the late 1880s and 1890s.
Even if Lincoln and Grant had each served three terms, Grant would have left office in 1885 (shortly before his death, as usual ignoring the Butterfly Effect).