‘As’ has multiple meanings. It can be used for stuff like comparisons, but also has this meaning related to time. In this context, it means “while this is happening”, so it gives a sense of continuation to the relevant action. ‘When’, on the other hand, tends to be more instantaneous. In “when you doze off”, dozing off is conceptualized as a binary state. Resulting from the moment you fall asleep, something else then happens. In “as you doze off”, dozing off is conceptualized as a gradual process, during which something else also gradually happens.

Yes. This is one of the meanings of 'as'. This is from Macmillan.

Yes, this is very common. You can use "as", "when" or "while" as synonyms in the example you give. They can all have the meaning of "during the time that". Be careful though because there are plenty of cases where you can't just substitute these three words, one for another. In general though, "as" can always be used when there are two events happening at the same time.