1. Yes. EIA chip component codes.
2. Ball Grid Array. JFGI
3. Check a dictionary. "Kitted" is, you bought enough parts to assemble the boards (plus extras), and you send them a kit. "Turnkey" is, they order the parts themselves (with nominal convenience and overage costs added in, you see).
4. Lead free is required to sell in Europe. There may be a cost difference too. Quote both and see!
5. Conformal coating is goo applied over the board, after assembly, to protect against corrosion and stuff. JFGI for more info.
6. Stencil is for placing solder paste prior to assembly (pick and place) and soldering (reflow).
7. Assembly testing can be done automatically (flying probe). Functional testing would probably require you to create, and send to them, a testing plan, which they have an employee perform. You may need to supply equipment to do this (e.g., programming dongles and software, custom support devices, test fixtures), or they probably have facilities to create these things themselves (in which case, they have to add design cost; so, this would only be something you would do for production testing, when you don't have in-house test building experience).
Tim