Poproszę herbatę - Tea please (you use POproszę when you ask to be given something. I'd use this one or the one that Bzibzioh mentioned)
Proszę herbatę - Tea please (this one is also ok, but the previous one sounds better imo)
Mam ochotę na herbatę - I feel like drinking tea (kind of...)
Chcę herbatę - I want tea (demanding)
Chciałabym herbatę - I'd like some tea
Would you answer differently if you were in a restaurant setting as opposed to at a friend's house?
no, these expresions can be used in both situations