I wouldn't really call them nanobots; they don't function in any active manner. They're really just like their digital counterparts--malicious program data that gets executed by our cellular machinery.
They are not pure code. They come with a protein coat and a bunch of proteins/enzymes to bring the code to the cell or nucleus, and sometimes more functions.
Sure, but those are produced by the cell at the behest of the viral code; not produced by the virus. The freshly-minted copy of the virus doesn't get to react to stimuli by producing proteins once it has left its host cell. It's a lot more like the relationship between, say, pollen spores and the gametophytes they contain.
To put it another way--a post-code and stamp on an envelope will get the postal system to deliver its contents to the right destination. That doesn't mean the contents are functioning to deliver themselves there; they're relying on a working mail-transport system that wants to take them there.