Learn Polish: Jem wszystko oprócz mięsa. - I eat everything except meat.

The preposition oprócz (except/besides) always forces the noun that follows it into the Genitive case. Because mięso is a neuter noun ending in -o, its Genitive form is made by changing the -o to an -a, giving us mięsa.

I've seen oprócz translated as 'besides' or 'in addition to'. How do I know if it means 'except' or 'in addition to' here?

You have to rely on context. Oprócz is a versatile preposition that can mean both exclusion ('except') and inclusion ('besides/along with'). In this sentence, common sense tells us it means 'except meat' (describing a diet), but structurally, Polish uses the exact same word for both meanings.

What case is wszystko in? Does it not change after the verb jem?

It is in the Accusative case, because it is the direct object of the action (what you are eating). However, wszystko is grammatically neuter, and for neuter nouns and pronouns, the Accusative form is identical to the Nominative (dictionary) form. So it doesn't look like it changed, but grammatically it is in a different case!

Since the current lesson also teaches zamiast, how would the sentence change if I used that instead?

Structurally, it wouldn't change at all! Like oprócz, the preposition zamiast (instead of) also requires the Genitive case. You could say Jem wszystko zamiast mięsa, which means 'I eat everything instead of meat,' though it sounds a little funny in real life compared to eating something specific instead of meat.