...is there a mathematical model that predicts when a signal sounds like steady noise, and when it sounds merely rough or dissonant?
Roughness is a quantifiable concept in Psychoacoustics. It is measured in aspers and the most commonly used method of measuring it is due to Aures (see also references at the second article I am linking above) which attempts to generalise Roughness as defined over a single tone to multiple bands.
But, Roughness characterises loudness fluctuations and I am not sure how much it may cover all the examples you are mentioning (?)
Another thing you might want to try is a direct quantification of the spectrum with metrics such as Shannon's Entropy as derived by the amplitude spectrum or the entropy of the phase spectrum. In other words, you would be assessing the "coherence" of the sound. This does not involve perception, but it would be sensitive to whether the waveform starts sounding like noise and as a quantity, it could be used in regression in case you are trying to use it to drive some sort of decision about the composition of the generated spectrum.
A significant unknown here is the number of components the sound is made up from. If you have 3 tones at random intervals between them, you get a different type of "dissonant" than if you had 81 tones. The potential for noise-like waveforms is higher in the second case. So there might be a limit of "density" beyond which the perceptual and structural metrics do not deviate much and you might be able to assess the structure of the sound with the simpler metrics I mention above.
Hope this helps.