It's not possible to do with just flash, you'll need to use either Flash Media Server (costly) or Red5 (need a big of java experience, but open source) in conjunction with flash to achieve your goals. Using either of the above will allow you to record, playback, pause etc. To save the file locally you'll either have to make your app an AIR app, or use PHP or similar to grab the .flv file from the server and save it to the user's HDD.
Has your soundcard got the option of recording "What you hear" or something similar?That's my preferred method to record the output of the soundcard while playing something with sound recorder on windows, but not very professional. Besides, some other recording app can be applied to recording streaming music or any sound from your computer's speakers.
Yes you can do both for desktop (AIR) and web applications.