Teaching Calc 3/Analytic Geometry

I know this isn't exactly the sort of question usually asked here, but I was hoping someone could help anyway...

I'm a "TA" at my university, but a very good teacher, so I am actually teaching Calculus 3 this semester. We are using Calculus: Early Transcendentals (6e) by Stewart.

I am "supposed" to cover chapters 12-16. But I think I'm going too slow; I'm only in section 13.3 (Arc Length and Curvature) right now.

I have a few questions to ask on how the class should be taught...

- How much explanation would you give on definitions/theorems, in the way of proof, for example? Obviously I'm not proving every little thing (like properties of dot product for example lol...), but I really feel it is important for them to see the reasoning, the logic behind these things. I really don't like just stating something and moving on.
- Are there any sections that can/should be skipped? I'm thinking 13.4 (Velocity and Acceleration Vectors) can mostly be avoided... maybe a quick mention of the relationship , but definitely skip stuff like Kepler's laws of motion.

- Just looking ahead to chapter 14, Functions of 2 Variables and Partial Differentiation... How much detail is really necessary here? I mean limits, yes... the fact that you now have infinitely many paths on which to approach a point as opposed to just from the left or right. But in general proving that a particular limit DOES exist seems to be quite difficult, without resorting to the characterization (which I don't really want to spend time on...). But other than that, there are a ton of PAGES in chapter 14, but the material is not difficult. There's just a lot of it, and I don't want to continue falling further and further behind...

So... I don't know. What would you guys do?