"En jente er et barn."

I think one is the neuter, maybe?

It depends on the gender of the word. Technically there are three genders: feminine, masculine, and neutral, although the feminine is rarely used in bookmål and can be replaced with the masculine form. Some places replace the "ei" with "en", but replace the "-a" ending, it just depends on the dialect.

Feminine: "ei" / "-a" Example: "Ei ku" / "kua" Masculine: "en" / "-en" Example:"En stol" / "stolen" Neutral: "et" / "-et" Example: "Et hus" / "huset"

Er means "am, is" Et means "but" I think such

It's sort of like English with "A" and "An". It just sounds weird saying for example "An Man". It's more like "A man". Same with "An ant". One wouldn't say "A Ant". It just sounds weird, but in Norwegian it's "en", "et" and "ei". An example would be dialects in Norway, for example; "A girl" can be pronounced "Ei jente" or "En jente" depending where you are, but it doesn't matter much.

Yes, they are treated as synonyms.

Apparently, it is also related to Old High German which was also "barn", "bearn" was the Old English word which also gave rise to the verb "bear".
http://www.merriam-webster.com/dictionary/bairn http://www.oxforddictionaries.com/definition/english/bairn

'Et barn' means 'a child'; 'a baby' is 'et spedbarn'.