1. assuming that enginner is a technical thing, artist is artistic, and producer is marrying the two, and bringing third party focus. what would your ideal blend of a mixer look like (i. e. one part engineer, one part artist, two parts producer)
Three parts producer, one part artist, no engineer. Or just enough to be able to know how to make certain sounds- which is really producer-ville, I think.
2. as a mixer, is it better to go into a mix completely cold (never hearing the song or having artist input) or going in with a lot of input from the client?
The tracks are the input. They should say what's being intended themselves- at least they're going to say what is POSSIBLE never mind what's intended. If the tracks say Dragonforce and the artist thinks they are Mayhem, the artist is wrong and the mix needs to obey the tracks, not the artist, to reach an audience. Trying to push stuff away from what it is weakens it, and artists so often don't know what they really are.
3. as a mixer, is it better to work alone, or with clients in the room while you mix?
Alone. It takes too long, clients will go crazy and start micromanaging stuff like routing
4. assuming you have free reign on a mix, is it appropriate to add tracks should you feel it really makes a difference?
If you're the producer- and in THAT case the client better be in the room, not out of it. Ideally they should have specifically asked you to add a particular thing, yourself.
5. if you were to listen to your work as a third person, how would you describe your style as a mixer?
Competent with heavy vocals, usually trying to sound nice.
6. how do you, yourself, describe your style as a mixer?
Old school