Are you able to unlock trophies or at least make progress toward unlocking trophies in full games if you play the demo?
No. Demos are completely independent games in terms of software. Any progress toward a trophy you make in a demo will not apply to the full game. There are some exceptions in the form of full game trials. If you play the free version of a game for a while and then upgrade to the full version, trophy progress will carry over since it is the same game, but there are very few of these kinds of games. In 99% of cases, the game demo will not have any connection to any PS trophies.