Can I redeem OS X Lion on a Mac different to the intended install location? I restored my MacBook to factory settings and I bought OS X Lion because my iTunes account never had it.

However, I can't activate it because the Mac can't continue unless it has the program installed or says "you must sign in with an Apple ID that was used to purchase OS X Lion."

Can I I redeem my activation code on another Apple product that has the Mac App Store in my name? or will the program install onto that MacBook?
Yes you can. Since the redemption code is tied to an Apple ID and not a machine, you are free to redeem it on another Mac.