When I'm at home my phone auto switches to wifi. I can SMS as if it was still on data with nothing to configure....just sms away.
this is normal, the phone still uses carrier data to send the sms. is not really over wifi. put your phone on airplane mode then enable wifi while at home. Check if you will be able to send sms.
most likely not.
It is possible to do txt and MMS messages over wifi. I know this works on iPhone's on at&t and Galaxy S3 on Sprint. I took for granted that all carriers supported this on all devices but apparently not. Neither of my fiends who have a GS3 on Vz or at&t can do txt/MMS over wifi. My one friend on at&t was doing this on his iPhone just prior to converting to a GS3. It's easy to test this by putting your phone in airplane mode and then turning on wifi.