Is it possible to continue upgrading an Android device if the OEM stops supporting it? Suppose the OEM loses interest in my device; can I still install the stock distro updates when issued by Google? Are there issues with device drivers if I go this route?
Once a manufacturer abandons supporting the device, your only choice is rooting it, unlocking its bootloader (if possible) and installing custom ROMs.

Google only releases the Android source code into AOSP (Android Open Source Project,) and has no say whether or not the manufacturer or carrier will update a given device.  The only exception is the Nexus line of phones that are supported by Google directly.  As it stands, AOSP cannot be built for any non-Google device without major work.  Even though manufacturers and carriers are required to provide the source code, it is true only for the part of the OS.  They don't have to (and rarely do) provide sources for the proprietary drivers, which are necessary to build a completely working Android version for a given device.  The community developers usually use the drivers from older OS versions, which greatly limits the upgrade potential.  The drivers sometimes are reverse-engineered which gives a greater flexibility to developers.

There also is an issue of locked down bootloaders on most new Android phones and tablets, which limits the extent of the changes a custom ROM can bring.  Most manufacturers and carriers lock bootloaders to prevent people from "bricking" their devices.  There are ways to unlock/bypass such locks for many devices though.

Bottom line is that if you are looking for a device that will be supported the longest - your best choice is a Nexus line.  Otherwise, I suggest looking at various online forums (xda-developers or rootzwiki, for example) to see if there is enough interest in your device from community developers to produce custom ROMs for it.