You could easily put together your podcast without using a mixer.
Using any kind of multi-track recording software (and there are dozens, ranging from free to very expensive), you can build your podcast up in tracks. That's the way all recordings are done, whether it be songs or podcasts.
Other than the actual interview, there's nothing "live" about podcasts - they're a produced audio show. You need to take a bunch of components and piece them together into a finished production.
It seems to me that your basic track is the audio file that you've recorded from Skype. You'd import that into your software, and do some "cleanup" to remove all the ums, ahhs, mistakes, and glitches. This also gives you the opportunity to snip that audio file into pieces and either rearrange the pieces or put spaces between them for music or other segments.
You can then add other tracks one at a time, including music, sound effects, advertisements, spoken intros, etc. These other tracks "overlay" your original one. You can adjust the levels of each track - for example, you can bring in a music track, then "quiet it down" when it's supposed to be a background to spoken parts.
With regard to importing sound effects or other snippets, all software will allow you to import those in a variety of audio formats like wav, mp3, wma, ogg etc. After importing them, you can move them around on the timescale for that particular track to have them play at the appropriate time in the podcast.
You could easily send the project files back and forth to each other to work on the podcast, although it would make sense for one person to do all of the editing, in my mind.
