Things I have found important so far:

When is your deadline?
Do you already have hosting?
Do you already have a domain name?
Do you already have a logo?
Do you already have context copy?
Do you already have decent photos?
What other websites do you like that are similar?
Do you plan to sell anything directly on the site (ecommerce)?
What language(s) will the site be in?
Do you need an iPhone version of the site?
What is your target audience for the site?
Do you need help with web marketing?

Most importantly - What to you hope to achieve by having this site?

I feel tempted to ask for "what is your budget" but I'm not sure if
that just opens up the scope for low balling. I would rather
be the one to set the initial "anchor" figure for the negotiation.

Is there anything I am missing here?

I think it's fine to ask about their budget. I do. I think there are a few things that you could add:

Describe your business, including culture and goals.
How are you currently marketing your business?
How do you plan to market your website?
What is the primary purpose of your website? What do you want visitors to do?
Have you owned websites previously? How did they do? What was your development process and how successful was it?

Good tips.

The thing about asking for a budget is this.
If you walked into a used car lot looking for a car
would you straight up tell the dealer how much money
you had to spend? I don't know I would.
I might even be a bit put off by the question.

Great tips but... this is off topic but how do you guys usually find your clients? Do you guys do online advertising or go around town and look for businesses =\