Wear on ropes make them fuzzier and cause more friction. Wear on tube style belay devices causes them to get sharp at locations.

As for minium thickness, some manufacture specify a full range of minimum and maximum thickness regardless of rope type, some do that per rope type. Food for thought.

Example the petzl reverso 4; "Rope compatibility: - for half ropes (8 mm or greater diameter) and twin ropes (7.5 mm or greater diameter) and on single ropes (8.9 mm or greater diameter)"