Can you send pictures through the messenger? Then it's probably soomething wrong with the messenger app. Try downloading another one and see what happens.

No? Then your APN settings (the MMS ones) are probably wrong. They're carrier-dependent, so go to your carrier's store and see if they can set them up for you. (There are no parts, it's just a matter of pushing a few buttons.)

 

I assume that you are on Verizon.... I suggest calling Verizon technical support and explain the situation. Years ago I had a problem where my account needed some sort of modification in order to accept MMS messages.

As for replacement messaging apps, just to check if it's a software issue on the phone, you can try Handcent in the Play Store. You can also try installing the app Verizon Messages from the Play Store. It's actually not a terrible app. The only issue, for me, is that I believe that the Assist app drive mode that offers to speak your text messages that you receive while driving only works with the stock messaging app.

 

Are you always on wifi? I don't have this phone so I can't speak with authority, but it could be that you can only receive multimedia messages with 4G active, not over wifi. That was the case with my Galaxy Nexus. If you have always been on wifi, I would try turning wifi off, 4G on, and then see if you can receive one.

 

No, not for MMS messages sent and received. The data for messaging is not counted against your data cap.

You can have WiFi on, but you do need to have data enabled. So long as data is enabled, when an MMS messages is sent or received it will use LTE, 3G or even 1xRTT data to try to send/receive the message, and then disconnect from that data stream.