Q: Can a defective catalytic converter cause clutch failure?

To answer your question, a catalytic converter cannot cause clutch failure. You have a series of independent failures, which is not very common for vehicles with less than 100,000 miles. I would take your 2013 Chevrolet Cruze back to the dealership and give them the opportunity to fix your clutch. You have only 16,000 miles on the clutch and this is clearly a very early clutch failure. Keep in mind, this could be another issue aside from the clutch. It sounds more like clutch hydraulics and not the actual clutch itself.

Your shift in 6th gear light will be an electrical issue. Again, this is most likely a separate problem. The P0420 code is another separate issue as well. Any of the following: "Engine misfire or running condition, large vacuum leak, engine oil leakage into exhaust-valve-guide seals, piston rings" could be causing this code. Although, I doubt there is an oil burning issue with such a low mileage engine. I’m referring to "engine oil leakage and piston rings." These can cause the P0420 code, but it wouldn’t be normal on a vehicle with less than 100,000 miles.

I recommend having your Check Engine Light and clutch inspected by a certified mechanic in order to diagnose the problem and make the appropriate repairs. Since someone else just replaced your clutch, it may be under warranty with them. This will depend on the source of the failure. With a low clutch pedal is likely not the clutch itself, but the hydraulic components that disengage the clutch so you can change gears. Good luck!