After getting the Crown of Barenziah I realised that I find flawless gems very easily. Are these two events connected or is it a coincidence? And if the crown has something to do with my luck, what do the other unique items do?

Getting the Crown of Barenziah gives you the Prowler's Profit effect which makes gems of all kinds appear MUCH more often in the game world. Pretty much every lootable object, that has a chance of holding a gem, will almost always have at least one. This means chests, burial urns etc... You will become rich very easily now and most likely be overburdened with jewellery.

As far as I know, the other trophies won't give you any special effects though.

Right, Ruhuratas, the other "trophies" you gather for Devlin only give you the gold that Devlin gives for finding them