Yes, yes it can
No; you have to have some sort of signal.
Here's the test: Turn on airplane mode on your phone, this will cut off all signals. Then, turn on wifi. Send a text. My text immediately got bounced saying "unable to send message".
You can however send texts through Google Voice; if you have an account setup.
I'm on WiFi 75% of the time and text just fine. When you're in airplane mode you are killing ALL radios even if you toggle them on. Airplane mode is like Operator OFF in a Siemens control system...It trumps everything. If you're referring to satellite, it will still work.
No, you're wrong. You can turn on airplane mode but still have wifi connected (you just have to toggle the wifi back on since the intial "airplane mode" toggle turns it off). This effectively kills your cell phone and data (3g/4g) connection, but still allows you to connect to a wifi network.
I'm doing it right now.
And attempting to text kicks back an undeliverable message while doing this, which tells me you can't text over wifi. But I can still surf the web and facebook and everything. Can't make phone calls or text.
EDIT: I should note I'm using the CyanogenMod ROM, in case Sense has something that prevents you from doing this. I will also note you could still use email to send a text message.
Oh hell...this is going to turn into the what freezes faster; hot or cold water thread...
If I'm on wifi, my 3 and4G radios are turned off...How is that I can text my other phone and receive it instantly? It can't switch at the speed I'm getting my texts. I see what you're saying about the web working...bizarre.
IIRC, SMS uses the the regular carrier signal, not 3G/4G (this is how and why "dumb" phones can still send text messages). So even though you're on wifi, SMS is still just going to use the plain old carrier signal.
And yes, I'm on wifi about 90% of the time, but I don't think that matters for sending SMS. You still need a carrier signal for it since it's not being sent over the wifi connection.