This is all from memory, so do your own research to confirm first please.
Early 6L80E (2007-2010?) TECHMs have to be matched with the same year engine ECM. Later years can be mixed and matched to work together with flash programming like EFILive or HPTuners. Later years (2010+) can also be flashed for a lot faster/snappier shift timing. Complaints from corvette owners about 6l80 shift timing compared to BMW DSG dual clutch shift timing. This resulted in a redesign of the shifting tables and programming that drastically improved performance with no changes to hardware.
To get tap shift working, with the stock computers, there are a things needed.
A resistors with specific values, a momentary switch setup and then flash programming the TECHM to enable it. It's not too difficult to find the corvette 6l80 wiring schematic showing the resistor values if you search around. Some people have even pulled the Malub SS/Cobalt SS console shifters with the tap up/tap down buttons and retro fitted them. Going this route can eliminate the need of a gm column shifter or third party switches.