No, this is an unacceptable swap. If you wish to go to a 50 series tire you should install 225/50-16 which is and excellent swap.
Okay, the "tyre number" is 205 55 16 (not 2055516). 205mm wide, 112.75mm from wheel to tread (55% of 205) and the center is 16" wide for wheels. So yes, 16" wheels will fit on your audi a3 and they will fit inside your 205 55 16 tires.
Replace the tire
The best idea to replace a tire which has a whole in it is to replace it with a new one. Reason as to why you should replace it because the tire can wear out and blow out sooner with a whole in it.
Yes , you can replace a 205 50r 16 with a 205 60 r 216. You must replace in pairs, not a single tire.
Yes you can
Generally requires removing and dismounting the tire, replace TPMS sensor, remount tire, balance tire, re-install tire to vehicle, do the relearn tire position procedure.
You can not replace just the tire pressure sensor on a Boxster. You will need to replace the whole sensor. The sensor is located inside the tire assembly.
225/50-16 is as close to a 205/55-16 as you can get. A 195/60-16 is also fairly close and is an acceptable replacement. My advice is stick with the OEM size that your car was designed around. There is no benefit whatsoever to going with any other size.
Yes, it is possible.
Yes, you can.
Yes you can
Yes you can