Firstly my advice, purchase the update to Mountain Lion (you can go from Snow Leopard 10.6.8 to 10.8), it's cheap, really cheap.
The only way to determine if the installer is genuine is to run a checksum of the contents, or download it as new which will require a purchase.
Off course it's impossible to know what revision of the installer you have, so this answer may be factually wrong but the process is valid.
If you have the installer for OS x 10.7.0 then this command
/sbin/md5 "/Applications/Install Mac OS X Lion.app/Contents/SharedSupport/InstallESD.dmg"
will return a checksum of
b5d3753c62bfb69866e94dca9336a44a; note this does assume that the installer is the /Applications directory and not somewhere else.
Can you use it to install and then install Xcode? Literally yes you can, but as the comment's say, if you don't own it, it's illegal for you to use there are however no technical restrictions on installation that will stop you, only your own moral compass.
You may however find that installing Lion updates from the Mac App Store will not work.
This all leaves you in a position where your Mac is less secure, cannot install essential security updates, and that is even assuming the Lion installer provided is not already compromised in some way.
If you have multiple machines in your house, then yes you can purchase once and run the installer on both machines. This is explicitly allowed in the Mac App Store terms and conditions:
(i) to download, install, use, and run for personal, non-commercial use, one (1) copy of the Apple Software directly on each Apple-branded computer running OS X Mountain Lion, OS X Lion or OS X Snow Leopard (“Mac computer”) that you own or control.