How to tell if new MacBook Air is a lemon?

Question

I purchased a MacBook air two weeks ago and have been noticing problems with it since then. I'd like to list them below and solicit your feedback as to whether this is pointing to a larger problem I should seek Apple Care help for, or if this may be something others are experienced and will possibly addressed by a firmware update.
Computer details:
MacBook Air 13"
1.7 ghz intel core processor
4 GB memory
purchased October 2011
Running OS 10.7.2
The first sign something was wrong was the battery life. I charged the computer, unplugged it, and the estimated battery time was 3:30 minutes (approximately). From the first charge.
The second sign something was wrong was when I shut the computer down with 52% battery remaining. The next morning when I started the computer it was dead. The computer will not start without being plugged in as the battery is consistently drained if it's put to sleep or shut down when not connected to a power source.
Third sign - the computer does not seem to turn off. This could be why the battery drained as mentioned above. When I have come back to use the computer an hour or so after I've shut it down (plugged in or in battery mode) the screen is gray and the shut down circle is still processing. This is a consistent issue.
I force shut the computer down by holding the power key, but when I start it the next day the computer flashes awake and asks me if I really want to shut down the computer, as if the force shut down had not even occured. The battery is not fully charged at this point (despite being consistently connected to a power source).
I think that is all. What does this sound like to you guys? This is my 6th mac laptop (first in 1995) so I'm used to different problems over the years, but this is my first time with issues like this one, and my first MacBook Air.
Finally, I brought the computer from the US to a European country, so cannot go back to the store where I purchased it to ask these questions in person. I'd have to switch my account to Apple Europe I guess. I hope that's possible.

Answer 1

See if you can narrow the problem through a process of elimination. First, run a Hardware Test to ascertain if the MBA has any issues that may be the cause or contributory to tthe problem.
To run Apple Hardware Test from Lion Internet Recovery on a MacBook Air (Mid 2011)
Beginning with the MacBook Air (Mid 2011) you can remotely start up your computer from an Internet-based Apple Hardware Test. These steps require a DHCP Internet connection, either wired (using the USB to Ethernet adapter, sold separately) or wireless:
Press and hold the Command and D keys while the computer starts up.
Release the keys when the Apple logo appears.
If you are using a Wi-Fi connection, select a wireless network from the menu that appears.
Follow the onscreen instructions to test built-in hardware.
If Apple Hardware Test detects an issue, it displays an error code. Make a note of the code before pursuing support options. If Apple Hardware Test doesn't detect a hardware issue, the issue may be software related.
If the system passes the Hardware Test, then move to software. You may want to consider a fresh format and re-install. You cam always cone the existing installation to an external disk and/or make a Time Machine backup, to restore your MBA's existing image, whichever way the re-installation proves to anarrow the cause.
If the fresh install still exhibits the problem, you should proceed with a warranty resolution to the problem. Then use the clone image afterwards to restore your system. If the problems disappear, then you know the prior installation had some fault. You may want to migrate the apps and data from the TM install and see if that works, or choose to reinstall your apps and just pull the pertainent data off the backups.