Sie can be used as a way to put distance between you and the other person. For example, among younger people du is common even between strangers (and is often used inappropriately to customers by poorly mannered service personnel!!), but Sie is still the correct "default" in most situations.

However, if someone has pissed me off, I'm much more likely to engage them with "Sie" because I don't feel they deserve the friendliness that du can also convey.

With regards to your other question, I think it depends very much on your level of German as to whether you are aware of it. The basic answer is yes - these things are important, but you will find that people put far less emotion into their speech and gestures than say in Italian.

I'm of the opinion that all of the subtle things we are aware of in English as natives are also present in German, but in the beginning many people may have a different impression. I'm referring to tone, intonation, facial expressions, body language, non-direct speech. German does tend to avoid some of the frivolous over-politeness found in some flavours of English (would it be too much trouble if I could possibly bother your for...). Though a little bit of additional politeness never hurts either.

Just a final point - as a rule Sie is not used to mean "naff off" (though it can be used this way as I stated at the beginning) it is simply a way of showing respect.
In English if I have to stop a stranger in the street for some reason I always say "Excuse me Sir". I use Sir to show respect to a stranger. Since English doesn't have the equivalent of du/Sie, that's the closest example I can think of.

Hope that answers your question. Next time give us a hint as to your level of German and it will help others to give you appropriate information