That very well could be the TV or more likely the settings on the TV...I connected my MBP, 15", i7 running Lion 10.7.5, to a Samsung LCD TV this weekend. Went to the Apple store and bought the Kanex Mini-DP to HDMI Cable, Part Number H4637ZM/A ($44.95) and the Apple Mini Displayport to VGA Adapter-INT, Part Number MB572Z/B ($29.00). Both worked with the MBP and Samsung. I had to go to the Samsung on-screen menu to select the input source for each, HDMI and PC so it would "see" the MBP. On the MBP, I opened System Preferences > Displays and the displays window showed three buttons top center...middle button was Arrangements which allowed mirroring to be used. Also slected resolution for the VGA to be 1980xwhatever it was for the TV. Both gave very sharp images using Keynote iWork presentation software.
The source selection is necessary on the TV, but I do not understand why your MBP is not giving you the Arrangements button in the Displays window...there is a breakdown in communication there, the MBP is not recognizing the TV. That should occur irrespective of the settings on the TV.
Does your TV support VGA input? If so, can you find someone to lend you the Mini-Display to VGA adapter so you can connect that way and see if the two communicate?
I doubt the issue is my use of Lion vs your use of Mountain Lion, but since I also have ML on the MBP I may just go and reboot into ML and try it that way to make sure the issue is not ML itself.