Anglo-Saxon refers to British people, but Saxony is in Germany. Are they related?

Sort of. In the post-Roman period, the British Isles were invaded/colonised by a bunch of tribes from modern north Germany and Denmark called the Angles, Saxons and Jutes. The native Romano-Britons seem to have referred to them all collectively as the "Sais" (a word derived from "saxon" which survives in the Welsh word for English, "Saesneg").

Other sources put the Angles first, for example in the name of England, which derives from "Angle-Land", and the poor Jutes seem to have been forgotten entirely.

Edit: Sorry, the modern areas of Germany called "Sachsen", "Sachsen-Anhalt" and "Niedersachsen" are of varying relation to the historical Saxons - the state of Lower Saxony (Niedersachsen) is more or less where the Saxon tribes lived that settled in Britain, other Saxons lived in Saxony-Anhalt, but the modern state of Saxony, in the southern part of the former East Germany along the Czech border seems to have got its name from the title "Duke of the Saxons" being passed around various people in Charlemagne's day and ended up there.

Is the term "Britain" related to "Briton?"

"Briton" is to "Britain" as "American" is to "America". Annoyingly, they're pronounced the same.

How does it relate to Brittany, which is in France?

This one is slightly contentious - there are some who contend that it is "Little Britain" to the island's "Great Britain", but that's debated. Whatever the detail, it appears to be the case that it was named by outsiders in reference to the number of settlers from the British tribes who moved across the channel to what had been known to the Romans as "Armorica".

How did Britain have a house of Normandy, if Normandy is in France?

In 1066, Duke William of Normandy sailed across the English Channel and defeated the Saxon king Harold Godwinson to claim the throne of England. He and his successors are called the House of Normandy, and the king list was "reset" with William I, The Conqueror, so that when a new King Edward came around he was Edward I, despite there having been a couple of Edwards in Anglo-Saxon days.

Is Mercia related to Murcia, Spain?

Not as far as we know. Mercia in England is a latinisation of an old english word for "border", which is also where we get the word "marches" for a border area. The origins of Murcia in Spain are a little fuzzier but it's thought it relates to the myrtle plant.