Headphones serve several purposes...
We used them back in the old days where every detector was effectively like an AT Pro in all metal Pro mode. There was a very steady and faint threshold tone that we listened to while detecting and very deep coins or objects would cause a very, very faint increase in tone which would not be noticeable without headphones. The headphones served to be able to hear those faint changes in tone, but also served to block out external noise.
Using headphones also increased battery life as the power draw from headphones was less than that used to power the detector's internal speaker. Today's internal speakers are much more energy efficient.
While headphones do insulate you from noise and diminish our ability to detect people approaching us, you can actually compromise by using a very light pair of headphones that rest on the ear as opposed to covering them. You'll hear the tones from the detector and be able to sense and hear things/people around you. Besides, lighter headphones retain less heat and having them sit on your ear as opposed to around your ear will keep you cooler.
As for the quality of the tones, expensive phones may make the tones sound better, but I don't believe you'll miss anything by having a cheap pair of headphones.
As for wireless, there are a couple options out there. but they are very pricey. I'm betting there will be bluetooth transmitters on detectors soon, so you could even use some kind of MP3 player or phone with bluetooth and headphones as a receiver, or just use dedicated bluetooth headphones. I'm there when THAT happens. I can't wait to cut the cord!
- Muddyshoes