Is it customary to pay the full week's rental on a cottage 6 weeks before arrival?
Customary, maybe not. Unusual, no. I'd make sure you have some sort of travel insurance to cover yourself in case you have to cancel between now and then.
I agree with Maureen that it's not unusual, but if you have qualms check TripAdvisor to see if that property is reviewed and if there were problems. Travel cancelation insurance is a good idea, too. But paying ahead relieves you of the need to get a large amount of local cash immediately on arrival, and it's kind of nice to have that part of your trip paid for already.