Does Touch ID erase the device after failed tries? I don't have an iPhone 5S, but I know that I can set my iOS devices to automatically erase themselves, when the wrong passcode has been entered too many times. Does Touch ID on the 5S do the same? 

I imagine this to be rather inconvenient, since accidentally touching the ID sensor is very likely.
Quoting from About Touch ID Security, from Apple’s website:


  Touch ID only allows five unsuccessful fingerprint match attempts before you must enter your passcode, and you can’t proceed until doing so.


You then get ten attempts at your passcode before the device erases, but these are counted separately from attempts to authenticate with Touch ID.

This is slightly different from the other answer:


After three attempts, the iPhone will present the passcode screen, but you can still have two more tries at authenticating with Touch ID.
After five attempts, Touch ID is disabled until you successfully enter your passcode.


Apple’s security documents don’t mention a setting to erase the device after failed Touch ID attempts, and I’m not aware of one. I don’t believe that iOS devices would erase themselves after five unsuccessful attempts, at least not with the current version of iOS.