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"max-width" is preferable, so as to accommodate smaller windows.
Yes. Especially for mobile browsers. I created a site recently using the 960 grid system. This also uses (as the name suggests) a fixed width of 960px. Using this on a WVGA (800x480) mobile device is not really that usable.
So, as suggested, `max-width` is much better as it allows smaller screens to "squeeze" the page together, while it will simple "max-out" at the defined size on larger resolutions.