is σύζυγος an epicene noun that depending on the article refers to a man (ο σύζυγος) or to a woman (η σύζυγος)?
is its 'proper' accent σύζυγος? Because I think the second (in the last syllabe) is for του. So if we would have The wife drinks milk, would we have η σύζυγος πίνει γάλα?
1) Yes, that is correct.
2) The accent moves from the first syllable in the nominative case to the second in the genitive case. Ο/Η σύζυγος, του/της συζύγου.
You can use wiktionary to check grammatical gender and declensions, also for verb conjugation. While not perfect, there's a fair amount of content in wiktionary beyond definitions.
I don't know if it is possible to say in English "my woman" for "my wife", maybe it is slang, but I think (talking from my ignorance of Modern Greek) that the difference is like in Spanish (Spain):
mi mujer would be η γυναίκα μου
mi esposa would be η σύζυγος μου
In the case of Spanish, you can use both for refering to your wife; for example, the formula used in weddings is "yo os declaro marido y mujer" (not esposa). Although both are common, I think esposa would be more formal. Maybe in Greek is more or less the same.
I hope it helps, but let's wait for the answer of a native speaker :) .
I didn't know that's how it works in Spanish (that flag next to my username means very little really), but you're right, in Modern Greek both are used to denote the exact same thing. Ο/Η σύζυγος is a bit more formal so in casual conversation you would use άντρας/γυναίκα respectively.
Neither; it's [ɣala], but the sound [ɣ] doesn't exist in English.
It's similar to the "r" in many dialects of French or German, however. And it's the voiced version of [χ] which you may know from Spanish "j, g". (Actually, I think Spanish might have [ɣ] as an allophone of "g" in words such as "agua".)