[2019] Is there or will there be a search function and question on digging down

Yes! I do plan on adding a search to the build menu.

As for digging, you can change the z level you are on with Q and E. You can then assign a vertical mining job to build a tunnel. Stairs cannot stack but ladders can. Ladders are slightly slower than stairs but provide vertical pathing, so you could put them in this new vertical mined out tunnel. Alternatively, you could build a spiral staircase. Just keep in mind staircases need to be built on platforms.