Not all, not even most from the listings I looked at a few years ago. Only one of the three old Android phones I have supports it, and it's one without a recognizable brand (the other two are Samsung and LG).
No he or she is not, a radio receiver is not such a high tech after all and I think all of my mobile phones had or have that feature build in, even though I never used it (in some you would have to have earphones plugged in as a antenna)
I haven't had a phone with built in radio, but back in the late 0s I had an ipod with a built in FM radio, and it was the coolest radio I ever had. You could pause and rewind it like a tivo. I assume the radio on a smart phone would be about the same.