MacbookAir4,2 intermittent freezing/crashing. Lion re-install left system unusable

Question

This post is mainly to catalog my experiences with my Mid-2011 Macbook Air. At this stage it's looking like I cannot solve the problems I'm having.
Within the first week of purchasing the laptop the system started freezing intermittently. I the symptoms seem to match those discussed elsewhere on this forum (1, 2, 3). At the worst times the machine would freeze 4-5 times a day, but it could also go 3-4 weeks without any issues.
I called Apple support who told me to Reset the PRAM/NVRAM and the SMC. This did not solve the issue. I re-installed Lion (without formatting the HD), no change. I performed the Hardware test (standard and extended), it showed a clean bill of health. I upgraded to 10.7.1 and 10.7.2 without any fixes.
I have just tried to reinstall Lion after a format of the HD and appear to have put the laptop in an unserviceable state. After formatting the drive I tried to ensure I could choose it as a startup disk. It wasn't showing in the Recovery Mode Startup disk selector tool. So I bless'd the mount, it wouldn't set the NVRAM boot parameter. bless'd the device, same thing. I went ahead with a re-install, and it downloaded the image and restarted.
Now I have a    prohibition symbol showing when the Mac starts. If I try to use the Recovery tool (Command-R) it asks for my wireless password, and won't accept the WPA2 key (I've tried multiple times, and can access the WiFi from my Android phone, it's the right key). Trying to reset the PRAM/NVRAM, SMC and entering the Hardware test are now all failing. The recovery USB key I made shows a flashing folder with a question mark when I hold option and select it. I remade the recovery USB key on a MacBook Pro and that's showing the same thing (though Apple do say:
If the computer shipped with Lion, the external recovery drive can only be used with the system that created it.
It's got 3 year Apple care, so it will be brought to an Authorized Reseller in Dublin tomorrow who does Apple Care repairs, there are no Apple stores in Ireland.
I'm disapointed that Apple are still beta testing their products on users.
Addational information:
Model: A1369 EMC 2469 - MC965LL/A

Answer 1

I got my MacBook Air back on Tuesday. It has a new Logic Board. It hasn't crashed since the board has been replaced.
However, it now has two new problems: Wi-Fi & the Mini DisplayPort.
WiFi first. I'm not the only person with this problem. This was the 802.11 (Wi-Fi) adapter specs from System Information:
Card Type:     AirPort Extreme  (0x14E4, 0xE9)
Firmware Version: Broadcom BCM43xx 1.0 (5.100.98.75.18)
A ping to my home router looks like this:
That's when I can actually associate with an access point. I could not connect to the AP in work, a Motorola Netopia Model 2247-62 with WPA2.
I re-installed Lion, no luck. I followed Maayan's guide to downgrading my wireless drivers and I'm now able to connect to this Netopia AP, most of the time. Earlier it would not connect and console showed the following:
Dec  1 15:38:48 localhost kernel[0]: AirPort_Brcm43224: Ethernet address 04:0c:xx:xx:xx:xx
Dec  1 15:38:48 localhost kernel[0]: AirPort: Link Down on en0. Reason 8 (Disassociated because station leaving).
Without rebooting this problem seems to have fixed itself.
Ping responses from the router are better, I'm only getting ~7% packet loss:
For reference, this is not a problem with the router, my friends MacBook Pro shows the following ping response over the same time period, at the same distance from the router:
The monitor I tried using was a Dell E171FP. When plugged in OSX Display preferences can see it. It worked for ~10 minutes and then went into power save mode. After unplugging/connecting and rebooting it remains in sleep mode.
I have tried plugging the monitor into another Mac (same VGA-MDP adapter) and the monitor works with no issue. I've turned off display sleep in Energy Saver preferences, no joy. I'll try another monitor later.
Minor (and humorous) side point, OSX no longer knows what my Serial Number is: