Did Turkey do the 'right' thing by shooting the Russian jet?

Difficult to say. This was not an out of the blue incident, Russia has been constantly entering Turkish airspace. Turkey has shot down Syrian jets before, so Russia knew they were poking a hornets nest. That being said the jet was not in Turkish airspace for long, mere seconds. It also seems likely that this had more to do with Russia hitting nearby Turkish allies and so this is more of a message showing Turkish displeasure at that.

Looking at the situation in isolation I cannot help but feel the responsibility lies with Russia for antagonising Turkey. As part of the wider situation though? This seems to have been more a shot across the bows. Turkey is becoming something of a liability for NATO.

According to reports they entered Turkish airspace for 17 seconds and the jet was shot in Syrian airspace

Indeed, but I think the time spent in Turkish airspace is irrelevant given the circumstances. The fact that they were there at all is enough given that Russia had previously been warned that any infiltrations would be responded to by force.