Your questions are good, and essential. I would lean on the second one a bit harder -- it's important from a marketing standpoint that the client's corporate identity is maintained across media (except when it's so awful that you have to redo that from scratch, too).
Always ask who their competitors are, and why they regard them as competition. That will tell you a lot about what works in their market and about the client.
Find out what has been successful for them in the past, either on the web or in print. By all means get any brochures, ads or sell sheets, but if you can identify "that really successful ad we ran in '05" you'll be doing them, and yourself, a favor. It's amazing how often I'll find a client had a really successful campaign going and dropped it, only to be mystified a year later when sales were off.
I spend as much time as necessary at the beginning of a project getting into the client's head. There's a great video clip of an interview with Paula Scher on the Adobe website that you should absolutely watch, that perfectly summarizes the process. The end point of this conversation (or several) is you can look at the project completely from the client's point of view as well as your own. Nine times out of ten your first design mock-up then lands in the ballpark.
In understanding the target audience, get very specific: age range, income level, interests, other places they go to buy similar products. Get it to the point where you can personify the audience ("Maisie, the 20-something office worker who's into the club scene, lives with a room-mate, loves the latest fashion but always buys at discount stores because she's always cash-strapped. Loves YouTube.") This helps in determining the look, colors, and attraction point of the landing page and/or home page. I find with small to medium size business clients that in many cases it also clarifies their own thinking on who they are trying to reach.
Clarify for yourself and for the client what the end result of a site visit is to be. It may be a purchase, but more usually it's a reach of some kind that will later turn into revenue directly or indirectly.
The client very often doesn't know what they want, beyond something as vague as "we need a new website" or "we need to be more attractive." It's your job in the initial meeting(s) to turn the nebulous idea into very specific design goals. If you come away unsure of what's wanted, write up what you got from the meeting, send it to the client and set up a phone conference or another face-to-face. Persist until you know what they need. I figure it's my job to make the client as successful as I can, so it's not all "what do you want?" -- sometimes there's a healthy dose of "here's what you need, and why you need it."
By all means bring your designer into the conversation early. That's what Skype, Adobe Connect and the conference button on your phone are for. Introduce him as "my partner" or "a colleague who will be working on this with us" if you don't want the client to know how you are outsourcing.
Get the initial wireframe or sketch and go over it with the client before the design is fully committed. That way, if you missed the mark you can find out about it early. Your second shot will usually be on the money.