Would I still use mehr if the englisch read, "Please, drink some more wine."?
Yes, like that: "Bitte trinken Sie etwas mehr Wein." or "Bitte trinken Sie noch mehr Wein." (=most literally) or "Bitte trinken Sie noch mehr Wein."
Yes, the "Sie" refers to the person(s) you are addressing to. "Sie" is used for the polite singular or plural.
If you are talking to friend you would use the collegial form: "Bitte, trinke (du) mehr Wein."
I would use the "du" just to emphasise the person you are talking to - or better his/her name. Especially if there are more people present to whom you could be talking.
(Or some more slang: "Komm, trink noch 'nen Schluck!" = "Come on, take another sip!")
There is no sie (lowercase) in this sentence.
There is a (capitalised) Sie -- that is because commands addressed to Sie have to include the pronoun Sie.
(Grammatically, it's because they use the subjunctive mood, not the imperative.)
"have some more wine" would be a much better translation; "drink more wine" sounds very creepy, and would not be used in a normal situation.