Mine displays HOME when I pull down the notifications window and only for a few seconds when in the lock screen. Other than that it doesn't display HOME from what I've noticed. I don't think you have to worry about the what is displayed as the carrier's name. Since you are able to make phone calls and received messages (both SMS and MMS?) and the only thing that is not working is just data, then it's probably something in the phone configuration or a problem in your area. If you haven't done so yet I would recommend contacting Straight Talk.
What symbol, if any, do you have to the left of the signal bar?
I know this probably isn't the case but did your turn Wifi off? If not try turning it on or off and making sure it stays off so you can connect to the data network.
I don't know if you have done so but you could try a factory reset, not sure if this would help.
If none of the above works you could try rooting and flashing the AT&T stock rom to the phone or some other rom. (Consult this recommendation with someone more experienced here on the forums. Personally, I don't see more troubleshooting options.)
If there is no problem with the network and none of the above works then I think your phone may be damaged, unless it used to work perfectly with AT&T. Did you switch from AT&T to Straight Talk or did you buy the buy through ebay or on the street to activate ti with them?