"Do F-mount autofocus lenses work in autofocus via the FTZ adapter when shooting video on the Z bodies?"
Yes, if they are AF-I, AF-S, or AF-P (or third party lenses with internal lens motors, such as Sigma's HSM lenses). AF-I and AF-S lenses will make clear noises that are audible and picked up with the camera's microphones; the AF-P lenses tend to be more silent, though can still be heard with the camera's microphones.
"Is the focus as fast with an FTZ-adapted lens on a Z body as it is on a DSLR?"
Not necessarily. With the Z9, yes, focus speed is as close to identical as with a DSLR as I and others can measure, and is often clearly faster than the DSLR when using a teleconverter on the lens. With the other full frame Z cameras, focus speed can be somewhat slower than with the DSLRs, though again with a teleconverter the Z bodies improve compared to the DSLR. The teleconverter change isn't surprising: with DSLRs, teleconverters put the light getting to the focus sensor at or near the minimum needed to focus at all, whereas the mirrorless cameras tend to be able to focus at speed no matter what the effective aperture is. Also, note that the Z System cameras tend to focus slightly more precisely than the DSLRs do, particularly for single-focus.
"Wait, what about screw-drive lenses?"
Older autofocus Nikkor (or third party) lenses that rely on a camera body motor to drive the focus system will not autofocus on the FTZ adapter. In general, the D-type lenses from the film era have screw drive systems and thus won't autofocus when mounted on a Z camera.
"Can I use AI and AI-S lenses on the FTZ adapter?"
Yes, and you can set exposure with these lenses. Obviously, they don't autofocus since they're all manual focus lenses ;~).
"Will we ever see a Z to F mount adapter to use the new Z lenses on DX and FX DSLR cameras?"
No. The Z lenses were designed to provide infinity focus when sitting 16mm forward of the sensor. The F-mount is over 46mm from the sensor. If you move a lens that far forward of where it was designed, it will lose infinity focus (and probably a lot of distance focus, too). If you wanted to use use a Z lens on an F-mount body as a macro lens, I suppose you could create an adapter that acts like an extension tube, but I don't see much demand for that.
"Is it possible to build a teleconverter into a Z adapter?"
Yes. It might be a little tricky to optimize, but I don't see why it couldn't be done. There's been no indication that Nikon is doing so, however.
"Does the FTZ adapter work with third party F-mount lenses?"
Information has been getting better over time about this. Recent Sigma (Art) and Tamron (G2) lenses are fine. Both those companies have issued firmware updates for recent lenses, and have actively written about compatibility on their Web sites. Older Sigma and Tamron lenses are where the issues start to pop up, and both companies seem to have a break point at which they won't do firmware updates. Basically, I'd say that anything pre-USB dock from either company is a hit or miss situation.
"Will the FTZ adapter make F-mount lenses perform optically worse?"
No. The Z7, for instance, uses pretty much the same technologies and coatings that the D850 uses. An adapted lens will be at the same position relative to the image sensor, so the light ways will hit that sensor the same. Thus, I see no differences between the way a lens performs in terms of image quality on a D850 natively versus on a Z7 via FTZ adapter.
"What happens with DX lenses?"
They work as expected on the FTZ adapter. The camera automatically crops to the DX format (and can't be overridden). The EVF compensates (e.g. the DX crop fills the image area).
"Does the FTZ Adapter maintain weather sealing?"
Yes with a footnote. At the camera end, the usual Nikon Z weather sealing is still applicable. At the F-mount end, it will depend upon the lens you mount on the adapter. Many older lenses don't have the rubber gasket seal that newer ones do. That said, generally I've not found the mount to be the place you have to worry about in inclement weather, except when you're changing lenses.
"Does VR on an F-mount lens still work?"
Yes, it does. And both the lens and sensor VR work in conjunction with one another. If the lens has a VR switch on it, that will control both the lens and the sensor VR.
