Any recommendations for reading the fault codes off my 1.4 TFSI? I thought I'd save myself some pennies and get to the bottom of what's causing the occasional loud beeping sound.

Get yourself a VAGCOM interface it's pretty close to dealer level diagnostic and can be used to read and clear faults program certain control modules reset service intervals graph live data and a host of other stuff. Have a look at https://www.ross-tech.com/ they only sell original interfaces be careful elsewhere because there are a lot of fake ones for sale online and these at best won't work or at worse could damage your car or damage one of its many control modules 

OBD 11