Dein = your (singular)

Euer = your (plural)

It depends on the number of people you are speaking to.

For the same reason that "Yours neighbour is noisy." is incorrect: you need a possessive determiner here (before a noun), not a possessive pronoun (which stands instead of a noun).

In English, you can say "Your neighbour is noisy" (with noun) or "Yours is noisy" (without noun), but not "Yours neighbour".

In German, the possessive determiners inflect like ein or kein, and so they have no gender/number/case ending for masculine nominative singular, neuter nominative singular, or neuter accusative singular.

Thus for Nachbar in this sentence (masculine nominative singular) it is euer Nachbar with no ending on the basic form euer-.

If you left out the noun, you could say Eurer ist laut. (Or Euerer ist laut. if you wish -- also correct, according to the standard, though it's much less common in my experience and actually sounds wrong to me.)

It could be either a female or a male neighbour, hence both Nachbar and Nachbarin are correct.

I suppose that's technically correct, and you can report it if you want.

But geräuschvoll is not a common German word, and your sentence sounds odd and unnatural to me. I'd recommend that you use laut.

• You misspelled Nachbar
• The possessive determiners mein, dein, sein etc. inflect like ein, kein -- meaning that they have no ending before a masculine singular noun in the nominative case. Thus it is Ihr Nachbar and not Ihrer Nachbar.

No idea. That's just how it is in German.

Possessive determiners (before a noun) inflect like kein or ein and don't have an ending in masculine nominative singular, neuter nominative singular, or neuter accusative singular (Mein Vater und mein Kind sehen dein Kind).