I can answer the first question: who would have become president if Andrew Johnson had been removed?

Current law dictates that the Speaker of the House of Representatives would become President if both the offices of President and Vice-President were vacated. However, this is due to several laws and Constitutional Amendments that weren't around in 1868. The applicable law at the time was The Presidential Succession Act of 1792. Section 9 states:

That in case of removal, death, resignation or inability both of the President and Vice President of the United States, the President of the Senate pro tempore, and in case there shall be no President of the Senate, then the Speaker of the House of Representatives, for the time being shall act as President of the United States until the disability be removed or a President shall be elected.

The President of the Senate pro tempore (who, under current law, is next in line after the Speaker of the House) is the presiding officer in the Senate when the Vice President (who is also the President of the Senate) isn't around to preside. In modern times, the President pro tem (the hip, short version of the title) is the most senior (longest serving) member from the majority party. In most of the 19th century, however, it was a position that a senator would be elected to by merit, or by favor.

So who was the President pro tem in 1868? Benjamin Franklin Wade, Republican Senator from Ohio, would have become President if Andrew Johnson had been convicted in his Senate trial. He was considered a "Radical Republican", which at the time meant that he did not look favorably on letting former Confederate states and officers off the hook for the Civil War. There is some conjecture that the unwelcome prospect of Wade becoming President was a deciding factor in the vote not to convict President Johnson, but I'll let the experts chime in there (it's something I happened to see in his Wikipedia article while I was researching the succession and pro tem stuff, so it's not exactly a well-sourced claim on my part).

Wade would not have been president for long, though. Section 10 of The Presidential Succession Act of 1792 states:

That whenever the offices of President and Vice President shall both become vacant, the Secretary of State shall forthwith cause a notification thereof to be made to the executive of every state, and shall also cause the same to be published in at least one of the newspapers printed in each state, specifying that electors of the President of the United States shall be appointed or chosen in the several states within thirty-four days preceding the first Wednesday in December then next ensuing: Provided, There shall be the space of two months between the date of such notification and the said first Wednesday in December, but if there shall not be the space of two months between the date of such notification and the first Wednesday in December; and if the term for which the President and Vice President last in office were elected shall not expire on the third day of March next ensuing, then the Secretary of State shall specify in the notification that the electors shall be appointed or chosen within thirty-four days preceding the first Wednesday in December in the year next ensuing, within which time the electors shall accordingly be appointed or chosen, and the electors shall meet and give their votes on the said first Wednesday in December, and the proceedings and duties of the said electors and others shall be pursuant to the directions prescribed in this act.

Now, 1868 was an election year, and the vote in the Senate to convict Johnson was in May. Three Articles of Impeachment were voted on; Article 11 on May 16, and Articles 2 and 3 on May 26. All three were decided by a 35-19 vote, 1 short of the 2/3 necessary to convict (it was the same group of 35 and 19 on all three votes). According to Section 9 of the Succession Act above, if I'm interpreting it correctly, there would be no need to have an election to replace Wade and finish out Johnson's original term, because the next regular election would take place within the proscribed timeframe ("within 34 days preceding the first Wednesday in December then next ensuing"). Inauguration of the next president would have been (and was) in March, 1869. So Wade would have served 10 months as the ultimate lame duck.