If the wall geometry is grouped, you need to open the group for editing (right click on it and choose Edit Group) and then draw the rectangle and push through.
Another common problem for newbies is that they don’t get things drawn on plane. This could also cause the problem but without seeing your model, it’s hard to tell if that’s an issue.
As for getting rid of the Z-fighting, you have to do things that don’t make it occur. Why did you draw the rectangle on the floor? If it is to become a cabinet, using Push/Pull to make it 3D will prevent that from showing. If you are trying to divide the floor into smaller sections, again, you need to open the group for editing before drawing the rectangle.
FWIW, I know that it can be exciting to see your model with textures applied but I would suggest you avoid adding textures early on. They can mask geometry problems like reversed faces and they can cause a performance hit as you are modeling. Save your texturing for after the geometry is pretty well complete.
If you share the .skp file we can probably give you more ideas.