> they have multiple distinct spoken languages sharing the same writing system
It's... complicated. If you sort of squint the writing system is maybe the same in the sense that across different Chinese varieties cognates are almost always written the same way (e.g. Cantonese and Mandarin). But a lot of non-prestige Chinese varieties don't really have a standardized writing systems (although a very interesting phenomenon is that speakers can generally tell you when a given word has an associated character(s) they could write down for you and when they can't think of one). So usually when a writer of a non-prestige Chinese variety writes she/he is writing in Mandarin, not in the variety itself.
Nonetheless, there have e.g. been historical Chinese novels written in non-prestige varieties (e.g. in Min or Wu) so in that sense yes the same writing system can and has been used, but different authors might use different characters depending on how they understand the relationship of certain words to the prestige variety.
Even the different languages part is complicated. https://news.ycombinator.com/item?id=16844074
As for what phonetic "spelling" out happens in Chinese, usually a loan character is used (e.g. if live transcribing something where the transcriber hears some unfamiliar words, the usual solution is to write down characters with the same sound and mention that this is purely a sound loan in parentheses).