RE: Record video from iPhone to Computer
I'm playing with this same issue -especially "hardwiring" the iPhone to the Mac to get the best bandwidth...
There is an AirPlay-like option but as you say, there is a limit to WiFi bandwidth so the results depend on what video quality you're looking for, and it also means you need a local WiFi network available which can be a challenge if you're shooting "on-location".
Anyways, disclaimers aside, there are two products that do a great job of making your Mac an AirPlay Server...
Reflector by airsquirrels.com and
AirServer by airserver.com.
So from your mobile device, you enable AirPlay with mirroring and you'll see the Mac as an available "server". It does a great job of displaying the mobile device onto the Mac's screen and both products also allow for recording the session. Granted, both products are meant to allow you to "project" the mobile devices so it's not meant for video recording but it should work. And these products are less than $20. Good luck.