Sync picture in picture clips in Davinci Resolve
I've seen how Davinci Resolve makes it very easy to create a Multicam
  clip and sync the angles. However that seems to work to show just one
  angle at a time, and I want to show both at the same time. How do I
  accomplish this?


It does only show one angle at a time.  The way you accomplish it is to create two tracks on the timeline, drop an instance of the multicam on V1, duplicate that instance by alt/option dragging upwards to V2 (add shift after you start dragging to constrain the movement vertically, and to keep the clips synced temporally).  Then switch the angle of either one of them, so that V1 is showing angle A and V2 is showing angle B, for instance. Then adjust the transform and crop parameters on the topmost clip (V2) to expose the one underneath.  

As for the bonus question, you can either be careful how you blade by making sure you have either both or neither clips selected when you hit command/control-B, or you can lasso-select the two clips, right-click, and select "Link Clips", then make sure clip linking is on by toggling the chain link icon (cmd/ctrl+shift+L).  Clip linking lets you move clips as groups, blade them as groups, and displays a red frame offset badge if the clips accidentally loose sync.  I'm not sure how lightworks works, but it's very similar to Premiere and FCP7.