I know that he is considered the author of the document, but was the final draft written by his hand or was a scribe hired to do the actual writing?
This question could also apply to any large handwritten document. I ask because comparing the rough drafts that were presumably actually written by Jefferson to the final draft I can see a significant difference in style.
Additionally, I would like to know how one would go about writing such a large document by hand. Were there any techniques at the time to correct errors, or would the author have to start again entirely if s/he made a mistake? Was it common for politicians and other important people to be trained extensively in penmanship?
Good pennmanship was considered an essential part of a good education until quite recently -- the decline of penmanship is really a 21st-century phenomenon.
Well educated men (and women) of the 18th and 19th centuries were expected to have good handwriting, especially since so much communication and official business was transacted via hand-written letters.
Scholars and scientists were further expected to have several "hands" -- ways of writing the letters, roughly equivalent to modern print ideas of "italic" vs "bold".
As it happens, the the "engrossed" (meaning official) copy of the United States Declaration of Independence was drafted by Timothy Matlack, a member of the Pennsylvania delegation. Matlack had a particularity good "roundhand", and the members of the continental congress called upon him several times to engross formal documents including Geo. Washington's 1775 letter of commission.