Does this thing have a sound chip that can be programmed like a basic synthesizer - ie attack/decay/waveform/etc - like the SID chip in the Commodore 64?

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Nope. Apple was specifically prohibited from doing this under the terms of their settlement with Apple Corps. The closest Apple ever came to something like this was the Apple IIGS which included an Ensoniq 5503 sound chip (leading to another lawsuit), and would be better suited for this kind of thing.

But for all the details about what the Classic II can do, check out this Apple Developer Note.