Yes you can, just plug in your headphones into the headphones socket on the TV. (if the cable is long enough)
Yeah, you can even get a headset that has a microphone that does what you want. Check out "Turtle Beach X11's". They should suit your needs.
Either can be used. I use sometimes an extension wire from the tv (about 5m one off ebay for £1.99) and some skull candy headphones.
Or you can get wireless gaming ones like the x31s (£35 off ebay) and they will work via a small box from the tv to the headphones . Or similary through the xbox.