Can a Macbook Air mid-2011 boot without any SSD installed?
You don't need an internal drive for a MacBook Air (any Mac for that matter) to boot; you can boot off an external drive.

You also don't need a battery to turn on a MacBook (plain, Air or Pro).  Plugging in the MagSafe will just show an amber led, not green.

It doesn't turn on because there's something more serious wrong with it.  It's most likely, the SSD was removed for two reasons:


Salvage the user's data
sell on the secondary market


There are many logic board repair services available, you may want to check them out because it's most likely a logic board issue.