It's not normal, especially if you have only a double crankset!
I've set up triples such that they have no noise in any combination. And that's cheap components. Think 8 speed bike with Shimano Altus FD, thick 6-8 speed chain.
I recently set up an incorrect front derailleur on an 8 speed bike, with its original shifter: a Shimano SLX "Dyna-Sys" FD for 10 speeds. Using a 9 speed chain and some tinkering, plus a very slight unorthodox rotation of the derailleur around the seat post away from perfectly parallel (it's a clamp-on, not direct mount), I got it so that it has no noise on the middle and small ring, any cog, and only a bit of chain rub in the largest-cog-largest-ring combination.
If mismatched hacks like this can be made to work, there can be no excuses from the bike shop selling you a brand new bike for over $1000, in which components should match, and be properly set up. They are not a department store; they should know what they are doing.
From the fact that you have a rub on the smallest cog, which progresses to the second smallest cog when you shift the FD to the left to the middle ring, suggests that the FD is positioned a little too far to the left: either it could use a little more cable tension. What you can do is relax the high/low trim screws and get it as good as you can on both rings by cable tension; then play with the trim screws.