It brings a wider image to the screen -- it makes the image true 16x9 widescreen, rather than 4x3. It uses the full surface of the CCD, rather than cropped-down letterboxed subset like squeeze and letterbox mode do. And the anamorphic also gathers an optically wider shot (it's a wide-angle adapter on the horizontal axis only) so it gives you a 33% wider shot with the full height of the chip.
It's harder to focus because it sets a different focal plane for the horizontal elements than it does for the vertical elements. You need to carefully balance your focal distance, aperture, and zoom setting to make sure you have enough depth-of-field to bring both the horizontals and the verticals into focus in the same shot.
1) its problems with focusing aren't relegated to getting everything on the screen to focus (i.e. if everything in shot is exact same distance from the camera, the camera doesn't have problems keeping all of that in focus) but rather it's D.O.F. focus it has problems with.
2) Does anamorphic descrease your D.O.F.? If so, by a lot, or just a little?
3) (I lied) I have heard people bitching and moaning about using anamorphic for focus problems and all that other nonsense in film as well. I thought that all these things you're pointing out that are necessary when shooting anamorphic (focal distance, aperture, zoom setting) are standard procedure in film regardless on whether you use anamorphic or not.
1) You have to experience it to understand it. Yes everything at the same distance will fall within the same in-focus or out-of-focus plane. However, again, the lens focuses on the vertical plane at a different distance than it focuses on the horizontal plane. It can be very difficult to judge when something is in proper focus -- you really can't do it just by eye, and it's not easy to do it by monitor either. You have to manage your aperture first and foremost, and if you're not at a small-enough aperture there is no way to get the subject to be in focus. And that proper aperture depends on how far the subject is away from you, and on what zoom setting the lens is at.
2) Not really. A tiny bit, but nothing to write home about.
3) Anamorphics in film cause other problems, because the squeeze ratio is so great (in film it's a 2:1 squeeze, in video it's just 1.33:1) so you frequently end up using lenses that are twice as long as you'd use for spherical production, which makes the already too-shallow 35mm DOF become absurdly incredibly too shallow, which makes holding objects in focus very challenging. And then anamorphic lenses are typically slower than sphericals, so you often have to open them up to max, which makes for even more difficulty in holding objects in focus... Which is probably one reason why the big movies seem to be embracing Super35 instead of anamorphic recently, and especially with the new theaters installing new screens that actually show a smaller 2.39:1 area, cropped down from the larger 1.85:1 area, it's really lessened some of the appeal of shooting anamorphic 35mm.
But those focus issues are not the same as what the Panasonic Anamorphic faces... it's not a matter of not enough DOF overall, it's a matter of having an astigmatic focus situation where you have to try to find a compromise that brings both the horizontals and the verticals into focus at the same plane.
The anamorphic adapter guide takes up like 30+ pages of the DVX Book, explaining how it does what it does and what you have to do to extract the optimum results from it.
The future anamorphic from Century should correct all problems, but you'll need to focus twice: once for the verticals and once for the horizontals (at least as I understand it). But you should be able to get razor-sharp focus at any aperture and any zoom setting with it.