Carpentry can’t find wood

The first tile you have pictured is a wall/pillar waiting to be constructed. Pending walls will have their colors inverted. The second set of tiles are completed walls. Pillar and Wall are synonyms and function the same. Pillars just happen to be where walls end. Pillar tiles will automatically become wall tiles if you extend the length of the wall.

I believe that core quality refers to the item itself - i.e., a superior-crafted door, and total quality takes into account the masterfully encrusted with blue diamonds and other miscellanea.

I've been using "Core Quality" for a furniture stockpile of masterwork doors (admittedly, in 31.16) and only masterwork doors are going to that pile.