William Burke did not confess to the crimes in order to save his friends and love, he was in fact betrayed by William Hare who sold him out after they were caught. However, this is comedy and Burke's confession was played for laughs.
Two hangings are shown in the film, in the style of a "long drop" designed to break the neck and cause instant death. This style was first introduced in the 1870's by William Marwood as a more humane approach. Before then, and certainly in 1828-1829 when the film was set, the victim was simply suspended by a rope around the neck, and choked to death by suffocation.
Burke and Hare are in Greyfriars Churchyard, digging up a body, but are watched by a dog on John Gray's grave; this is presumably Greyfriars Bobby. In reality, Greyfriars Bobby lived around 1855-1872 so would have lived later than 1828/29 when the series of murders took place.
Hare comments that his wife Lucky has "fallen off the wagon" when she resumes drinking alcohol after a period of abstinence. The idiom of "falling off the wagon" would not be in used in Scotland at this time. The phrase originated in America very close to the start of the 20th century.
When Burke and Hare stop for a drink and Burke says that they have no money, Hare plucks a half-crown from underneath Burke's cap. The half-crown that is used in the film is from 1902 - 1910 which is nearly 100 years after the film is set.
In the "multiple dissection" scene Tom Wilkinson uncovers each of the four dead bodies one at a time. But, following a quick edit to the applauding students, in a closer shot one of the bodies is covered up again.
When the photographer takes the picture of the cadaver in Doctor Robert Knox's premises, he removes the glass plate from the camera and exposes it to (low level) light in the room. That action would have spoiled the photograph and an image is visible on the photographic plate. This would not have been seen.