Answer to: Why doesn't my tv sound resume when I turn the headphones off.The headphone stand that is connected to your TV is the cause of your issue. As long as it is plugged into the TV, the TV will send the sound out to the headphones. The bad news is that this is pretty standard behavior for any device with a headphone jack. Some background: The headphone jack acts as the indicator that tells your TV that you want sound out the Headphone Jack versus the speakers. Inserting the cable into the jack tells the TV to turn off the speakers and send the sound to the headphones. And vice versa, Disconnecting the cable from the TV lets your TV know that you want sound back through your speakers and not your headphone jack. Unfortunately I'm not aware of any TV's on the market that would be able to change the sound from TV speakers to headphones simply by turning off the headphone switch on those headphones that you are using.
Answer to: What is this wire for?It's hard to see perfectly from the picture but that appears to be a piece of coaxial cable that connects the wifi module to the wifi antenna. The two solder points at location 1 are for the coax shield and the inner conductor itself. That "sticker" at Location 2 is the wifi antenna. Not sure what your options are, but I would unsolder at the board. Don't peel the antenna up. If it tears, your wifi reception will be diminished.