Your pickaxe won't prevent you from equipping other gear pieces, so if it is, it's a bug.

As for your second question, when you have the pickaxe on and you're a miner, make sure you have the Prospect buff on. Then you can use Lay of the Land to point you towards where a node is. Look in your gathering log to see zones. There are some nodes right outside the gate to Ul'dah in Western Thanalan, so try going there.

You won't see rocky outcrops, another kind of gathering node, until I think it's level 10, when you unlock your sledgehammer offhand.