How do I use Resolve to permanently extract and save small clips from mov files/
You can do it right in QuickTime.


Open the videos in QuickTime
Edit > Trim
Save or Export those shorter clips
Import them into your preferred editor


If you do want to use Resolve:


Import your clips to your Media Pool
Click on the Edit tab, select all of your clips in the media pool, drag them to your timeline.  The order of them doesn't matter, if you just want individual clips, neither does having gaps in your timeline.
Once you have trimmed your clips as desired, go to the Deliver tab
Make sure to select Individual Clips at the top and adjust your settings as desired 
Select your entire timeline and then add the job to your render queue
Render