For both Coupes and Roadsters, typical problems to be aware of include saggy leather seats, noisy / failing window regulators, knocking steering racks, broken hood opening lever and poor (ground) connections to rear lights. There's also the risk of a broken glove box hinge if and when the damper stiffens up.
A VCDS or OBDeleven will provide some information including chain stretch, but there are a few things you will need to physically check as well -
Be aware that 2.0T TSI engines produced between 2009-2011 are prone to high oil consumption. Be sure the owner has a full service history.
The high pressure fuel pump cam follower is another issue Audi failed to mention during routine maintenance. If the cam follower fails, it can destroy the cam and will be an expensive repair. It's recommended for replacement every 25,000km.
Hole worn through cam follower results in damage to the cam lobes.
If you are looking at a Quattro, be sure the Haldex oil and filter have been changed regularly. Many service shops, including Audi, don't replace the filter because it doesn't show up on the parts list. But the Gen 2 and Gen 4 Haldex have two filters, including a small strainer on the pump.
Haldex Filter and Strainer on the pump