I think of tazza as more like a mug or a teacup (something with a handle) and bicchiere as a glass, or a "glass" that is made of plastic (like you would be served a soft drink in)

No, exactly the same. They both mean "among 2 or more items" (I don't know if you can say that in English)

I translated "La nostra tazza e fra i piatti" and it was accepted.

Stare is the equivalent of the Spanish/Portuguese verb "Estar". This verb deals with location. Ex: Sto alla scuola.

I think it's because the verb is referring to position.

You always have to use the article before the possessive pronouns, except before noums that indicate a member of your family:"mio figlio"( my son), "mia sorela" (my sister).

http://italian.about.com/od/grammar/a/italian-preposition-tra.htm Someone gave us this site previously. It is excellent.

Nostre is plural form of nostra. Bicchiere, although it ends with an "e", it's a singular form of a masculine word, hence the use of "nostro" (and "i nostri bicchieri" in its plural form).

Nostre is feminin. Bicchiere is masculin. Then, " il nostrO bicchiere" is the correct form.