"A blue cup and two plates, please."
Translation:Синюю чашку и две тарелки, пожалуйста.
This sentence is basically a request with the verb like «дайте» 'give' or «принесите» 'bring' dropped. Since you would use accusative for an object of these verbs, you use it in the short form of the phrase too.
Yes. It can be either nominative plural or accusative plural. However, «си́нюю ча́шку» is accusative, so we can assume «две таре́лки» is accusative, too.
Oh! Indeed. You’re right. «Две» is accusative, «таре́лки» is genitive. But genitive singular and accusative plural look identical here.
I'm far from an expert in this, so don't rely on the specifics, but I think the concept will help clarify things for you. Genitive sing is used with 2,3,4 and larger numbers ending in 2,3,4, e.g. 42, 54, 93. (I believe it is a linguistic remnant from a time when quantities/cases had three categories: one was singular, two was considered "dual", and everything else was plural. In Russian and a few other languages they kept the "dual" category and added three and four to it. So any number w/ 2-4 at the end is genitive singular (or you can think of it as dual) and 5-9, and 0 are genitive plural.