They're rotating too.

If you rotate as part of some larger rotating thing then you still rotate. (You also move around.) It's all absolute.

Indeed. The Eiffel tower and your head do both have some (extremely small) centripetal force compensating for their rotation along with the earth.

(You can break that down in different ways, i.e. use various choices of generalised coordinates to describe it, so exactly what constitutes "centripetal", "centrifugal", "gravitational", "tidal", etc. forces depends on that. I'm being pretty vague in how I decribe it. Regardless, rotation is absolute, or in other words the equations of physics take a different form in a rotating frame of reference than in a non-rotating one.)