In other words, if I receive a MtGox code and redeem it to my account, can the person who gave me the bitcoins, via the code, see that my account was the account that redeemed it?
Nope. Or, at least, unless you are law enforcement and have a subpoena.
Or, unless you redeemed it yourself (which would list in your deposit history).
You can tell if it has been redeemed by redeeming it yourself and seeing if it had funds, but that's about it.