You're right, bvdd, those are all *excellent* questions which are often danced around but not too often addressed directly. Good post!
1. Yes, with caveats. Most monitors strive for "accurate" or "flat" reproduction within the constraints of cost. Obviously it's much easier to get a flat-responding monitor at the $800 price point than it is at the $300 price point.
That said, however, there have been popular and famous monitors in the past (and present) who were also well-known not to be that accurate or flat. My favorite example was the Yamaha NS10, which a few years ago you found in just about every control room worth it's salt. Response-wise...well...lets say they could probaly never get THX certification. But everybody had them because they were a "common denominator" de-facto standard; engineers knew just how they sounded, to it was easy to move from studio to studio and know pretty much what to expect.
These days, however, there seems to be a (thankful) trend to accuracy. Especially with the advent of 5.1 surround and the THX certification specification. There will *always* be some coloration, of course, but the idea is to get a speaker that sounds as accurate as possible given the price point/budget.
2. A "flat mix" is a different animal, however. If all mixes were flat, they'd all sound the same. On most mixes, one does want to fill out the spectrum; not emphasize one band too much or leave too many deep holes in the spectrum. But a "flat mix" will sound "flat", and that's not always a good thing. Bottom line: mix to sound good, not to "look" good.
Additionally, there is what's known as the "equalization curve". Simply stated, the higher the frequency, the less energy it requires to hit a certain volume. -6dB at 2kHz sounds louder than -6dB at 200Hz. By the true definition of "flat", a "flat-looking" mix would actually be shrill and tinny because there'd be too much high freq. A "flat sounding" mix has higher peaks in the bass ranges than in the treble.
3 & 4.) See #2 above for the answer.
HTH,
G.