Do the tuner nuts screw down enough to hold the tuner firmly to the headstock? The tuner nuts rarely screw down all the way to the base of the tuner itself. But as long as they screw down enough to hold the tuner firmly to the wood, then you're fine. You'll need to use a spanner (10mm) to get the tuner tight enough, but if you are using reasonable force on the spanner and the nut's stopped turning and it's still loose, then there's an issue.

The small screw is only there to stop the tuner rotating under string tension. The tuner is held on by the nut.

As Dozymuppet says, there is normally round plastic stuck in the rear of the tuner to protect the surface of the round disc as it's inserted during manufacture. This is meant to be peeled off.