HI,
The iPhone has both Messages (send iMessages to Apple items and SMS to non Apple phones) and FaceTime.
FaceTime is a 1-1 Video chat app to other iPhones or iPads with FaceTime and Apple computers that also have FaceTime (People with Snow Leopard can get it and it comes on Lion upwards).
iChat is a Mac Only text, Audio Only, Video Chat and Screen Sharing app.
It does not connect to FaceTime.
FaceTime on your iPhone and them using FaceTime on Mountain Lion should work.
Mirroring the iPhone to the TV via Apple TV (Airdrop ?) is not something I have done. (it's not the side of Apple Products I know so well).
8:28 pm Friday; November 8, 2013
iMac 2.5Ghz 5i 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad