Nothing from Abraham will be informal. He is not that kind of writer. It would be like cussing in a church to him. He can't bring himself to write in a relaxed way. When this man writes, he can only do it formally. Yet, he was not educated at a college either... I don't think. I do not think he went to high school.. His high level of vocabulary is due to when he lived. His sentence structure is also. Yet one thing you can not get from having conversations is punctuation.

It's very rare to find a text which is "perfectly" formal or informal, and a slight bending of the rules either way doesn't change the general tone. So no, using "but" in this way doesn't make the letter informal. While starting a sentence with "but" is something to be avoided in general writing, I think if Lincoln's letter joined the two sentences, then the meaning of the whole sentence would change slightly, ie. the words following "but" would gain a higher importance than the first part of the sentence. Really not the effect Lincoln wanted, I suppose, so starting the sentence with "But..." was probably one of his better options. Any other thoughts?

This is formal English, written by a brilliant proper person many generations ago. He was self educated, and read everything he could find to read, which was limited in that day. To early Americans and British, even if highly educated, rules of grammar, and systematic, structured teaching did not exist. English speakers brought their culture, habits of speaking, and old-world training with them, that was later changed by educators to create a structured method of formal education. With the help of Noah Webster and his dictionary and other books that helped nationalize our education system, we began following a course of correctness in speaking.

normally it would be considered informal, unless of course, that the writer is placing 'but' at the beginning of a sentence to stress something. So in this case, I think it can be considered quite formal.