If the computer shipped with a backup or install DVD or flash USB drive
with Lion 10.7 for that computer on it, then that would be the tool to use.
If the computer shipped with an OS X 10.5.x or 10.6 system on DVD,
and the intended method was for the original purchaser to get an
Apple USB superdrive for MacBook Air, (as Apple had one for it) that
disk and a means to install it would be required.
There is another way if you have a computer that has an optical drive
and the MB/Air can sometimes use that other computer + disc drive.
•MacBook Air (Late-2008 and Mid-2009)
How to 'Erase & Install' using Remote Install Mac OS X:
http://support.apple.com/kb/ht3263
If Lion were installed from the internet or an upgrade via the Mac App Store
and If the person installing it chose to make Lion install a Recovery Partition
or Recovery HD on the MacBook/Air hard disk drive, then you may be able to
boot the computer from it; restart with the Command and R keys held down.
You may have to call AppleSupport or AppleCare and ask if you can get a
replacement original system media; if the computer shipped with Lion, then
the device may have been a Flash USB key or boot copy on thumbdrive.
Or it may have had a DVD of the original system, if OS X 10.6 or before.
{While Apple Support OS X info says your MacBook Air model will be able
to run Mavericks OS X 10.9.2, the lack of system resources could make it
slow and balk at running applications or multi-tasking; & it can't see any
hardware upgrades. It is stuck at 2GB RAM, & limited HDD/SDD space.}
For a conversation with Apple directly (online or main phone support) you
would have to provide the MacBook/Air serial number and model identifier.
A replacement would cost a fee, and availability may vary depending on
the actual system media your computer used.
If the computer shipped with an OS X 10.5 Leopard system on backup DVD
you could instead get a retail Snow Leopard DVD; a better base system to
start over with.
Should you have made a good backup method to save your work, and also
an external enclosed hard disk drive to use to restore a working system, then
you would be hours ahead. A quality externally enclosed hard drive can be
used to house a bootable full system clone; a working backup. Preventative.
You may not be able to get a copy of Lion, if the computer did not ship with it
and if there is no Recovery partition or Recovery HD in the MB/Air; you can't
even download it from Internet Recovery, from what I've heard lately. However
given the other issues you say this computer has, the first thing to do would be
to try and save anything off the computer, because any effort now would likely
require the OS X and all drive content (SSD or HDD) to be erased or lost.
If you have access to an Apple Store within some driving distance, you should
see if they have a Genius bar, and schedule an appointment with them in
regard to these issues your MacBook/Air has going on. If there is no Apple Store
(official) you may have to consider paying a specialist independent Apple service
technician at a reseller location to do whatever tech work, save your files to an
external drive, and attempt to install a new system; at a service charge to you.
Some resellers or specialists may have a retail Snow Leopard 10.6 DVD on hand,
while some could install a test system after they'd restored basic function, and
then your totally wiped MacBook would be running clean install. Without DVDs.
So most of this would require you to be ready for the eventuality of needing to
reinstall or perform major system maintenance, ahead of the need; lacking that,
you will have to search out hands-on help, who has the correct software options.
And also one who can do a proper diagnostic to check for other damages, etc.
I've looked around a bit, to see there is a somewhat similar 'restore macbook air'
discussion from these ASC pages, that rounds out what someone suggested an
other owner to try... https://discussions.apple.com/thread/4974418?answerId=21800543022#21800543022
The method you tried to change the Admin password may have caused issues;
so that did not really help address the first (or second, third) set of troubles.
•MacBook Air (mid 2009, late 2008, or original)
Reinstalling applications, reset password, and
using Disk Utility, using the MacBook Air SuperDrive:
http://support.apple.com/kb/ht2735
There are OS X Utilities on the Recovery; Disk Utility, and other tools.
If you had an external enclosed HDD, you could get it ready to be used
as a backup, and see about using Carbon Copy Cloner to make a copy
(clone) on it. This is best done well ahead of emergency to learn how.
Well, I suggest you contact an expert, a specialist, or see if a Store genius
appointment could be made to help you get this MB/Air through this time.
Or if you know of an Apple User group in your area, someone may be able
to figure out the best way to get your MB/Air to work correctly again. The
unit should be tested, diagnostics are free at US & Canada Apple Stores.
Good luck & happy computing! 🙂