Neither Skype nor Audacity “play well with others” as you’re finding. If you do manage to force the recording to work, chances are good Skype will stop working. There are some people who just load the software, fire the whole thing up, record a podcast and go home. Those people are unicorns. Most people get what you got.
We recommend a dedicated Skype recorder such as Pamela paid software or one of the other offerings.
http://www.pamela.biz/en/products/
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/6
Pamela will record the far side and the near side as separate tracks in a stereo show to make post production filtering and effects easier, and it will do it in high quality WAV format to avoid MP3 compression distortion.
Koz