And as an aside, if you disconnect a pitot/static instrument from the pitot/static tubes, do these connectors auto-seal or do you have to put some kind of plug in them, to prevent false readings on other pitot/static instruments?
Pilot DAR already addressed this, but no, pitot-static systems aren't self sealing. If you do open the system, you also invalidate the IFR certification of the system, and a complete test and certification of the pitot-static system must be performed again before the aircraft can be certified once more for IFR flight.
To open these systems up requires mechanic certification (or "engineer" certification, if you will), and should be left only to those trained to work on the airframe. Improper handling of the components can introduce all kinds of problems. I've experienced a complete instrument loss on two occasions in airplanes in which someone had opened the pitot-static system and not capped the lines while work was being done. Insects entered the system, and died. In IMC, the insects apparently swelled and plugged the system off, causing all kinds of havoc. They were in various parts of the system, too, causing unusual failures that didn't conform to what one might normally expect from a blocked pitot tube, or blocked static source.
I was looking for an answer to my question in context of a failed VSI or maybe a backup altimeter. Suppose my VSI is showing a defect, say, reading +100 ft/sec while on the ground. I would normally note this, maybe tape a piece of paper over it to avoid confusion, and that's it. But now you're telling me that I should deactivate this instrument, which would require me to dive behind the panel and start messing about with the pitot/static system (actually the static only, but you get the point) - which I'm not allowed to do (and rightfully so, I'm not arguing that).
So even though a VSI is not required for VFR flight, if it shows a defect it has to be disconnected, not just placarded U/S. And I'm not allowed to disconnect it myself so in effect it has just turned into a no-go item, right?