So to answer your question, yes you can send and receive texts on your iPhone with your cellular data and wifi off but you cannot send and receive iMessages (the blue ones), these ones need your data access. If you turn off wifi then you lose the Internet, and if you turn of data, then you lose email capability.
Furthermore, can you send and receive texts using WIFI?
These SMS text apps run on your data plan. This means they can send and receive SMS Text messages over your WIFI signal. Smartphones will always utilize your WIFI connection to send data instead of the cellular network if it is available.
Can you text in Airplane mode with wifi?
Putting your phone into airplane mode turns off all its antennae - cellular, wifi, bluetooth, GPS. If you're in airplane mode, but have a wifi connection, you can send non-SMS texts over the wifi connection, using iMessage on an iPhone or various other messaging apps (WhatsApp, etc.)