As far as I know, Mandarin doesn't have multiple pronunciations for the same character-- does Cantonese? Aside of that, you could use ligatures for that, couldn't you?
Another large class class comes from vestiges of derivational morphology in Old Chinese: https://en.wikipedia.org/wiki/Homograph#In_Chinese
For instance, the character 度 in modern Mandarin can be pronounced dù (when used as a noun) or duó (when used as a verb), both of which derived from Old Chinese /daːɡs/ and /daːɡ/, respectively.
With Simplified Chinese characters, some of them come from the merger of originally different words that had similar, but not exactly the same pronunciations. For instance, both 髮 (fà) and 發 (fā) were merged into 发.
For the example you give, in Cantonese they're dou6 and dok6, respectively.
The 3/6 tones in Cantonese and ˋ (4th, falling tone) in Mandarin are the "departing" tone, which comes from the departing tone in Middle Chinese, which I believe comes from the -s ending in Old Chinese.
Yes, it did if you're not a mandarin speaker. The simplification process was biased towards Mandarin and there are some words that were merged that have different pronunciations in Cantonese but not in Mandarin.
Both do. A single, isolated Chinese character may have multiple unrelated meanings with some of them having an entirely unrelated pronunciation. It is, in fact, ubiquitous.
The idea is that each honzi has exactly one meaning is a misconception.
With respect to ligatures, if by that you mean the length of the same word across different Sinitic languages, that depends on the specific language and its phonology. Mandarin, for instance, has lost a large number of finals over the course of its evolution which has resulted in words generally being longer and requiring extra syllables to resolve the phonetic ambiguity. The Sinitic languages that have retained more finals (and sounds in general) tend to have more of shorter words. Cantonese is one of them albeit not the only one.
