"En gutt" = A boy. "Gutten" = The boy.

"En gutt" can be any random boy, but when saying "gutten" you are talking about a specific boy.

Yes, that's a thing in norwegian.

In Norwegian you put the definite article after the noun, not in front like in English, Spanish and French. The definite article here is "-en" which is placed after "gutt", but with neuter nouns it would be "-et". With feminine nouns it would be "-a". Sometimes, when there is an adjective before the noun, you will put "den" or "det" in front of the adjective as well.

You can say both, but it will often be pronounced differently depending on where in Norway you are.

You can say both, but dropping the e would make it less formal.