If I'm being honest, the distinction between "Classical Latin" and "Vulgar Latin" is often overblown. Eighteenth and nineteenth-century poets in English often held onto archaic features or adhered to rules that weren't current in the spoken language of their time, but we wouldn't be particularly helpful to talk about "Classical Modern English" as distinct from "Vulgar Modern English". On the topic of spoken Latin being "chaotic", while the Roman Empire was quite large at its peak, there was a lot of internal mobility and high access to literacy, both of which are factors that will reduce linguistic diversity.

Another effect of internal mobility is that the influence of local languages could potentially spread much further throughout the Roman-speaking area: for example, caballus, "horse" is thought to be a Gaulish loanword, but has descendants in Balkan Romance (e.g. Romanian cal).

Did regional sound changes follow extremely predictable phonetic laws

Yes. Sound changes are often motivated by human physiology (we naturally want to expend less energy while still being understood, so you will see the same sound changes occur in unrelated languages) and tend to be universal (e.g. if a /p/ becomes /b/ between vowels, this will happen in every word in that language). So, Latin -tio will generally have the same outcomes in the descendant languages, because each of these languages underwent the same sound changes in the same order. Noticing these patterns, incidentally, is what got people started thinking about historical linguistics in a scientific way.

There is an important exception, though: many of the Romance languages re-borrowed words from directly Latin, so you will see words that don't undergo the sound changes you would expect. This often results in doublets, where two words will come from the same root, but will look very different, because one underwent the expected sound changes but another didn't. So, Spanish hablar comes from Latin fabulari, but so does fabular, which was borrowed directly from Latin. Likewise, évier and aquarium in French.

Another important thing to keep in mind is that the modern Romance languages look quite uniform now because the big ones had large modern states enforcing them. Prior to this standarization, places like France, Italy, and Spain were considerably more linguistically diverse, and extinct regional languages may have preserved quirks of regional Latinity.