Turning cellular data= no text messages received?

If you want to send text messages to other iPhones while your cellular data is off, you need to force your iPhone to send texts as SMS instead of iMessage. Go to Settings/Messages and turn ON the "Send as SMS" option. Your iPhone will always prefer to send iMessage to other iOS devices, but this option allows texts to go out "normally" as SMS if the iMessage service is not available (i.e. if you turned off your cellular data).

Yes.

If your message/conversation thread is blue, you're messaging one another using iMessage (which only uses data) rather than standard SMS which appears in green (and relies on your texting plan).

iMessage is an entirely optional feature.

You can disable cellular data per application if you prefer, in Settings > Cellular > "Use Cellular For" --

and/or you can disable iMessage in Settings > Messages > iMessage, to reduce data usage but still text.

Nah, you don't have to download anything.

SMS simply means Short Messaging Service

MMS also means Multimedia Messaging Service

Both services are your standard 'texting' and they do not utilize your cellular 'data' plan.

That's the one. The green Messages app will use "i-Message" if it's enabled.