Is this a set phrase, meaning something along the lines of "dinner is served" or "dinner is ready"?
I have never heard anyone say this and I am a native speaker. "Cina este gata." would be for "Dinner is ready." or "Cina e servită." for "Dinner is served."
Could this also mean dinner is ripe and fresh? I thing an earlier lesson said that coapta was ripe.
Does Romanian commonly have adverbs? Is it different to say "freshly baked" as opposed to "baked and fresh"?
I think "The dinner is freshly baked" would be: "Cina este proaspăt coaptă".
No. In Romanian you must use the definite article, so "Cina (the dinner) este servită" is a correct sentence, and "Cină este servită" is a incorrect one.