Nope, the ColorPerfectDome™ can be used by all cameras that allow for custom white balance, including (but not limited to) Canon, Nikon, Pentax, Olympus, Fujifilm, Leica, and Sony.
Internal camera light meters are calibrated to middle (18%) gray, based on the Zone System from the early black and white film days. Having a consistent and universal tone for your camera to read as the target allows for ColorPerfect images.
No, you'll want to photograph the ColorPerfectDome™ under the exact lighting conditions in which you'll be photographing your subject. That means placing the target where your subject is to take the reflective white balance calibration image. Holding the dome over your lens will completely block out light and been ineffective.
It depends on your camera manufacturer. Sony has a small circular area within the frame that it samples the reference image from, so you do NOT need to "fill the frame". Other camera manufacturers, like Canon, do require as much of the frame as possible to be filled for your reference image to get the most accurate results.
Yes, you'll need to constantly be aware of light changing and re-calibrate every time your light changes. For example, if you're in a studio and swap out lights sources, modifiers, gels, etc., then the color temperature of your flash will change according to the new setup. Also, flashbulbs change color output over time, and tend to get "warmer" the more you use them. Never assume the white balance is going to be the same in studio from day to day.
The same rule applies if you're outdoors using ambient light. The color of light from the sun, clouds, reflections, etc., is changing constantly. If you change your location or turn your subject too far one way or the other, the light color has most definitely changed and needs tweaked.
Assuming all those images were photographed consecutively and in the same lighting, you can use the reference image in post-production by using the "Eye-dropper" tool of editing software to sample the ColorPerfectDome™ as your true Gray point, and then sync all the subsequent images to that initial sampled color correction.
LCD screens on cameras are a great tool, but they are always high quality, accurate color representations of an image. The color, brightness, tonal shift, contrast, and more, can all play a factor in how the preview displays on the back of the camera. Also, the color of your ambient light when viewing. The best guide you can use to check your work is to review the color histogram on the reference image to get a sense of your accuracy.
There are several variables involved here, including the type of file (jpeg or RAW), the software you're importing to, as well as the color calibration and quality of your computer monitor. Just like the back of the camera, there are several ways of adjusting your monitor to get an accurate representation of your file. There are also things called color profiles and LUTs used by photographic printers. If you know you're going to be using the files for printing, then you'll want to acquire those profiles from your printer to best set up your monitor to match it. This will enable consistency from capture to output. If you're only using the images online, you could invest in a monitor calibration device. At the very least, you'll want to explore the color calibration software options already built into your computer like color temperature adjustments and native color profiles. You may also have the ability to choose a series of images that look "correct" to you, and the display will adjust to suit your specific preferences.