Ok, so LOTS of misinformation out there.
1. If you received an "unlock code", your phone will not unlock. Iphones don't have an unlock code. Unfortunately, Sprint support is horrible, just like all the rest of them. they give you misinformation all the time. Unlock of an Apple device comes from Apple itself. Sprint sends a request to unlock the device. When your phone connects to the Apple servers, it updates your phone, and allows the sim card to be unlock. There is no code to deal with. They gave you the MSL code, which is use for Android devices, not Apple devices.
2. Once Apple receives the request from Sprint to unlock the phone, your phone needs to communicate with the Apple servers. That can be done in many ways. The easiest, is to simply drop in your sim card, and the new software automatically searches and updates the carrier settings, and your phone simply works. This works about 90% of the time now days, but it might have something to do with which IOS you are on. If that doesn't work, you can put your sim card in, and plug into iTunes. You should get a message saying Congratulations your phone is unlocked. If not, the last resort is to do a complete restore of your phone from iTunes. At the end, you should get the congratulations.
yes, you can use a Sprint Iphone 5 on a GSM network, even in the USA. Sprint claims that the 4s and 5 are only unlocked for International use, but that is simply not true. @Ethmoid, you mention the problem is using a CDMA device on a GSM network, but there is no problem there, as the Iphone 5 can handle both, as it has both radios. It simply needs a sim unlock. Maybe you were thinking something else.
The bottom line is, that it sounds like your phone is NOT unlocked, due to the fact that you received a code from Sprint. This is not an unlock. You need to go back to them and demand they unlock your phone the "factory" way.