In 1860, Republican Abe Lincoln's running mate was Republican Hannibal Hamlin. But in 1864 Lincoln ran on the Union ticket with Democrat Andrew Johnson. I understand why but I don't understand how. Who made the decision to dump Hamlin and put Johnson on the ticket?
Is Lincoln responsible for picking Andrew Johnson? If not, who is?
Also, were John Wilkes Booth and the other plotters motivated by the opportunity to put a Tennessee Democrat into the Presidency? Did Abe Lincoln sign his own death warrant by putting Andrew Johnson in the Vice Presidency?
You said you know why, but I'll cover a bit of that as it helps tell the full story.
Lincoln did not choose his running mate. If you can believe it, back in the day party bosses had a lot more power than they do today. Before the turn of the 20th Century, the Presidential Candidates were chosen by caucuses rather than Primaries and they were different than the caucuses today, which are opened to the public. These were closed meetings of party bosses deciding on who should be the next candidate for President. Lincoln himself was chosen by party bosses as the best choice for the Republican party in 1860.
In 1864, with the end of the war in sight, the Republicans along with Northern Democrats who were in favor of the war called War Democrats (as opposed to Northern Democrats who did not approve of the war called Copperheads) formed a unity party called the National Union Party. In order to maintain a showing of that unity, the party selected Andrew Johnson, a War Democrat from Tennessee to be the Vice Presidential candidate. Hannibal Hamlin was a former Democrat, which made him enticing for the 1860 election, but the fact that Johnson had been a Democrat through the war was the real key. In fact, Johnson ended up switching back to the Democratic party after the National Union Party dissolved in 1868.
As for whether or not Booth was motivated in putting a Democrat from Tennessee in the White House, he was not. Or if he was, he had a funny way of showing it because the Assassination of Andrew Johnson was part of that plot. George Atzerodt was tasked with assassinating the Vice President, but he either got drunk and forgot or developed second thoughts depending on what sources you're using. Secretary William Seward was also supposed to be assassinated that night, but the assassin failed after being stopped by Seward's son. So, no John Wilkes Booth had no real motivation other than causing chaos and attacking the Executive Branch. Well, I can't say they were his only motivations, but he certainly wasn't thinking of getting a Southerner in the Presidency. At least not in the short term.
So I guess, to really answer your main question; it's the party bosses that picked him and Lincoln likely had little to no say, which is similar to other Presidents of the day. I don't want to do more research right now, but I believe the first President to choose his running mate was FDR when he selected Henry Wallace for his Third Term. Even then, I believe it was him trying to strong arm the party leadership into getting his pick rather than an actual official pick. Henry Wallace... now there is an interesting figure. Look him up. We'd be likely living in a much different world if he was VP for FDR's fourth term.