Should not have installed gutter flashing/drip edge over older - on a reroof they should have stripped down to bare sheathing - they got lazy, but other than premature rusting possibility due to old flashing, probably no damage done assuming the ice and water shield overlays the top surface of the flashing.
Putting the new gutter edge between fascia and old flashing is right, so any runoff drops into the gutter.
There are two thoughts on exposed fascia board - some like tucking the gutter edge tight up into the bottom of the flashing so no splash can hit the fascia board, others like me like the gutter as low as it can go while still having the gutter flashing drip into the gutter, so there is more room for visible icing and icicle formation before it fills the gutter unseen, and to expose the icicles to sunlight so they can melt through and break off at the shingle edge. Leaving some fascia showing also leaves it open to sun and breeze so it does not retain moisture in a confined area, so there should always be at least 1/2" or more gap at the top of the gutter edge.
Under normal conditions, unless you are subject to extremely heavy rains frequently, the amount of splash onthe fascia board should not be a problem.