Does the iPhone 4 have an FM tuner?

No.

No.

Does the iPhone 4 have a built in DVD player?

Yes, but only for tiny dvd's.

Actually it does have an FM tuner built into the Bluetooth chip, but it isn't written into the software and thus not usable. I don't even know if it has an aerial connected, or if it would use the BT aerial.

It's actually an advanced Blu-ray player with record capabilities & Netflix streaming, although there is no indication Apple has any plans to turn it on via software update.

Cool! I can't find the DVD player app though, just the FM radio app. Gah 😟 looks like I will have to read the specifications page. http://www.apple.com/iphone/specs.html

Yes.

Just googled a bit. The Bluetooth/WiFi chip in the iPhone 4 is a Broadcom BCM4329FKUBG that has a built-in FM receiver/transmitter. Technically, it would perhaps be possible to receive FM broadcasts if only Apple would include the software for it. BT and WiFi share the same antenna, and so would FM too, if it only were written into the software.

http://www.ifixit.com/Teardown/iPhone-4-Teardown/3130/3