Skype shouldn't be playing back your mic audio. Just click the microphone icon at the bottom-right under the preview window. It will still capture incoming audio from Skype though (from whoever you're talking to).
No, there is no way to record on a per-process level; that isn't how the Windows audio subsystem works. Stop using Skype, focus on interacting with your stream/viewers. Casters who ignore their stream in favor of a personal conversation with a friend are snubbing their viewers, and will be less likely to get return viewership. If it's a conversation about the game, people generally want to hear what's going on at the teamwork level.