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The 2010 MBA's were only shipped with Snow Leopard (10.6), Lion was not shipped with MBA's until the 2011 MBA was released. Snow Leopard is not available to install by internet recovery, that could be the problem with it not being available.

I see that you end up trying to get Lion with internet recovery, but the App Store says it is not available. Lion was not available for the 2010 MBA unless it was purchased from Apple. Have you previously purchased Lion?
 
OP:

I suggest that you boot to internet recovery, and install ANY COPY OF THE OS that you "can get installed". I don't care which one that internet recovery offers you... just "take it" for the time being.

The idea here is to get the MBP bootable and running again.

Get there first.
Then worry about "getting further"...
 
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