Removing edges also removing face

Provided that all of the geometry is in one plane you can select everything and context-menu > Intersect with selection.
This should split any faces with crossing edges.
If all else fails try zooming in, and drawing over a thicker profile edge to see if that splits the face…

I suspect you are confused about how SketchUp groups operate. The “circle” (actually an ellipse) is grouped. That prevents its contents from interacting with edges or faces from other contexts. The rest of the model is “loose” edges and faces in the model context. You can’t alter the contents of a group without opening it for edit, but you can see other things through it. The edge you are deleting is actually from the rest of the model, and without it the rectangle ceases to exist because it is not a closed polygon. Below I hide the group with the circle and then redraw an edge to close the rectangle. Then I unhide the group and you can see the new edge showing through the group. Then I erase it again, and poof, there goes the rectangle!