Hi, and first of all welcome.
My Wife has a 116i M-Sport and I drive a GTI so I should be well qualified to give you an appraisal of both...
- Bluetooth Streaming - very good. I stream from Spotify no problems and have full use of steering wheel controls when streaming. No album art when streaming unlike the BMW (we have Professional Media) but you can plug the phone/iPod in and get the album art that way. IMO, it's not quite as good as the Pro Media in the BMW but still excellent and easy to figure out.
- Sound system - To be honest, I think the standard system is excellent in the GTI. We only have the standard speakers in the BM but the quality is woeful, and I've never heard the HK setup but all I can say is that I listen to band type music (Indie Rock, Classic Rock, Blues, Country Rock etc...) and I'm consistently impressed by the sound of the standard speakers in the Golf. It's much louder than the BMW system and right across the frequency range, it sounds superb with brilliant definition between instruments.
- Can't help with mpg, although I have a DSG and I see long term average of 38mpg. It's about the same as the Wife's 116i M-Sport (ZF8 Auto). In fact, on a long run my GTI sees 44mpg and her 116i about 41mpg.
- Diamond cut alloys can be repaired but there is a limit. They remove (cut away) the finish (similar to skimming a cylinder head) and therefore there is a limit to what can be achieved.
One thing I can add is that the level of refinement and quality of interior materials is superior in the Golf than the BMW. The clunk of the boot shutting is where you'll see it the most. The ride quality and steering is also nicer in the VW. That said, I do like driving the BMW and I feel the driving position is better. The media system is also far superior in the BMW.
Hope this helps!
Jon