[mst]

If your lenses are Ai-S lenses, you should be fine. If they are even older, carefully check compatibility first. Some old F-mount lenses can damage modern cameras.

As I understand it, with a Nikon camera (D7000 or higher) I can use the aperture ring in aperture priority  and in Manual mode, but am still able to use wide-open metering.

Ai-S lenses should work on all current Nikon DSLRs, but on bodies below the D7000 you can't tell the camera what lens is mounted (via the menu). Metering is limited then. 

Can I also use modes in which the camera sets the aperture, i.e., shutter priority and program?

No. You set the aperture with the aperture ring and the camera closes the aperture to the selected value when the shutter button is pressed. This is a mechanical link, so shutter priority and P won't work. On no Nikon. You need CPU lenses to do that.

Another question: I have seen that KatzEye is not selling manual focusing screens for most of the current cameras. Do you know of any other supplier for these screens?

 
See the link boris posted, you can find similar offers on eBay. However, I can't comment on the quality. Focus screens need to be manufactured very precisely. A minor error in thickness will lead to front- or backfocus. This happened to me even with a Katzeye screen in the D300.

I think a D7100 screen is already available, but D600 and D800 screens won't happen, because they cannot be easily changed.

However, feel free to send an email to Katzeye with any questions you might have. Rachael Katz usually answers within a day or two.

I don't think liveview is a usable substitute, unless a tripod is involved. However, on Nikon cameras the digital rangefinder (the "green dot" and two arrows in the viewfinder) works with anything you can mont on the camera, including non-CPU lenses.

-- Markus