How can I install Mac OS X 10.7.5 on my Macbook Air?
The MacBook Air should have come with either a DVD or a USB pen-drive containing the operating system installer. If you haven't got these in your eBay purchase, try contacting the seller to see if he'll send you a digital or physical copy. Otherwise you can try to borrow from a friend.

If you have a USB pen-drive with the installer, you can ofcourse install directly from there.

If you have a DVD with the OS X installer, you can use a USB-connected DVD-drive to run the installer. If you haven't got such a drive, you can use the DVD-drive on a seperate PC (or Mac). Apple has a guide for that scenario here:

https://support.apple.com/en-us/HT3263#

The easiest way to solve your problems seems to be to take the Mac to an Apple Store. They should be able to reload the OS on your MacBook Air. I don't know if the original USB pen-drives are still sold at Apple Store as they were back in the day.