The outsourcing company I am working with wants me to invoice the clients directly and they want me to give them a list of my prices. They want to know my general hourly rate, and they want a breakdown of what I charge for services (e.g., custom graphics, custom animations, forms pages, etc.). I'm really not sure how to approach this.

What I would recommend is researching prices on the Internet. Visit Google and narrow down your search to web designers in Minnesota. Compare. The norm is anywhere from $30 to $200 per hour, so it is highly variable. An hourly rate between $50 and $100 is competitive. Charging by the hour is good for us designers but less attractive to prospective clients. Plus, it's hard to gauge how much the site will cost the client unless you are very good at predicting how many hours the project will take you beforehand. You'll see all kinds of pricing structures. The key is finding the one that works best for you and is fair to the marketplace. If you overprice yourself, it will be hard to get clients. If you under price, you will be doing yourself a disservice.

How is it possible to use domain/server space bought from one vendor with a domain name registered under another?

Transferring your domain to a new Web Host means pointing your Domain Name System (DNS) to a name server that supports name-to-address translation. (Your Web Host should be able to help you with this.) It usually takes anywhere from 2 to 4 days before the process is completed. During this time, your website will be sitting on two servers—your old ISP's and your new ISP's. To know when the DNS has propagated, I recommend making a small modification to your home page (e.g., change the mouse over color of your links). This way, you will know when it's safe to cancel your old Web hosting service.