How can I extrude this middle part of the roof down so that I can start putting tiles on it? If I extrude + scale down it looks weird, same if I try extrude + move. I simply want it to become thinner, but not scale inwards on the sides.
Try Alt+E > Extrude along Normals. Not perfect, but it will probably do the job in this case with minimum geometry and a bit of cleanup afterwards.
Alternatively, you could separate the faces using P to create a new object, which you will be able to move freely. Downside is that you would have to adjust the polygons and create a new face for each side, of course.