Using ANR aviation headset with a PC
The answer to all of our questions is broadly yes, but to break it down: 

This company makes mic's for the Bose ANR audio headsets (quiet comfort line) that may fit your use case. 

You should be able to use a step up adapter to plug any standard aviation headset into 3.5 mm. It might be worth noting that many of the headsets out there now have bluetooth that you can bus audio through including the mic audio for talking on the phone. However its worth mentioning that most aviation ANR headsets are bulkier than your typical ANR headphone as they also offer passive isolation should the batteries die. 

AA batteries are the chosen option as they can be easily sourced just about anywhere and quickly replaced. Some people may chose to use rechargeable ones. I would think the lack of battery options has more to do with the fact that for a situation beyond regular old batteries many people may opt for the panel power variants.

FWIW the audio quality (for music) is decent but not headphone quality on my zulu 3 as most aviation headsets are designed to work well in the voice range and not necessarily the full frequency spectrum range that a set of headphones will work well in.