Is it possible, say with Lion recovery partition or any other way, to access individual files on the hard drive to copy them before reformatting it to do a clean install?
It is not possible to boot Mac OS in "any flavor" as it always fails loading kernel – including safe mode or single user.
It is an early MacBook Air, so it is not possible to use "Target Disk" mode.
Best Answer
You can boot your system from a system installed on an external drive. If your internal drive isn't damaged it will mount and you can copy files off of it.
If you have a Lion recovery partition you can use Disk Utility to "image" the main partition of your drive onto an external device before you format and reinstall. You will have to restore this image onto another device before you can pull files off of it.
It is arguably eqsier reinstall Lion directly on top of the existing OS which should create a bootable system without overwriting your files–but to be safe, make a backup image before doing this.
Insert mandatory lecture about the importance of a good backup strategy which should make emergency measures like this largely unnecessary.