Start and stop capturing while video is playing
There are two main way I can think of to do this, the first, which requires a little work after the fact, but certainly won't interrupt your play back, is to set marker points as you watch through the timeline.  Put the entire video in the timeline and then use the M key to place timeline markers at each point you wish to make a cut.  After watching through, you can place razor cuts on each marker and ripple delete every other section.

The other option, which I'm not sure if it will work without interrupting playback or not, is to set in and out points using shortcut keys.  I will set an in point, O will set an outpoint and Ctrl-K to add the edit to the timeline.  If that does end up interrupting playback, there is one other option that might work indirectly.

You could also try using a multi-cam edit with nothing on the second track.  This would let you monitor what is on your main playback and you could cut to and away from it whenever you like.  At the end you would just have to remove the empty space and get rid of the other track.