A simple formula will work OK for some products. See the free WoundRollSizer from AbbottApp. You can improve on this by back-calculating an effective in-roll caliper using a known case and then using the formula for new cases. The ‘help’ button will take you to a screen that will show you how. However, there is no single formula for predicting diameter accurately for compressible products. The best we can do is model the system and that is literally at the Ph.D. level. My thesis was the first to use roll growth (diameter) measurements to back calculate tightness using winding models. Alternatively, some high end winding programs ‘learn’ about roll build by monitoring the length and diameter of rolls just made to adjust certain parameters to make the next one closer to a target.
However, even if you had such a Ph.D. model, you still would not be able to calculate diameter accurately. That is because the inputs required are not easily measured. Here I am not just talking about Poisson's ratio and stack modulus, that require special setups. I am also talking about more basic things like length that is affected by the (unknown) tension and unknown MD modulus at the point of measurement. I am also talking about the thickness that is poorly controlled on many machines. I am also talking about moduli that are affected by formation that changes with time. Finally I also need to mention that tightness is also poorly controlled on many winders. In short, it is for all practical purposes impossible to do anything but fudge simple calculations to get approximate diameter as long as tightness doesn't change much.
The second part of your question is quite easy. Simple roll density (weight/volume) is extremely sensitive to tightness and quite easy to measure.
These concepts and most other must-know aspects are covered in detail in my Web101 Web and Winding Courses such as coming up in August 23-26 in Patterson NJ or as video-on-demand.