Continuity mistake: When George (dressed as Blackadder) is in the cellar and about to go upstairs, he has one cup on his tray, however when he arrives upstairs there are two cups on the tray.
Deliberate mistake: There are several historical dates wrong in this episode. For a start, Prince George didn't become Prince Regent until 1811, and yet this is set before the Battle of Trafalgar in 1805 (as Black Adder gives Wellington some advice on it). Similarly, Wellington didn't get the title "Duke of Wellington" until 1814. Again inconsistent with being pre-Trafalgar.
Deliberate mistake: There are a lot of jokes with the name of the book "The Prince and the Pauper." However the book was published in 1881, 51 years after the Prince Regent died.
Factual error: When Blackadder has disguised himself as the Prince and is speaking to Wellington, Wellington says, "I have with me here the most recent briefs from my General in the field". Blackadder, deliberately misunderstanding, replies "If you would like to pop them in the laundry basket". Blackadder is referring to the short, tight style of men's modern underwear, known as briefs, which were not invented until the 1930s.
Answer: The closest we ever get to knowing his first name is in series 3, when he claims it's "Sod Off". Blackadder asks him in series 3 where this came from, and he replies "Well, when I was a kid, living in the gutter, I'd go up to the other kids and say 'Hello, I'm Baldrick'. And they'd say, 'Yes, we know. Sod off, Baldrick.'" The S in this series is likely a reference to that, but it's not addressed, and given this Baldrick is clearly different from the series 3 Baldrick, we can't draw a direct link.
Shay