Answer to "What is the best glue to attach the Headband Cushion to the headband?"A great place to get help on headphones is https://www.head-fi.org/ forums, and GameFun8 http://bit.ly/2EhIEr8. Offf the top of my head so to speak!
Answer to "Using headphones in cold weather causes them to shutdown completely."Batteries do not like cold weather ( electrons don't like to move), especially if there are reaching their end-of-life. How old the headphones?