1.- Is “tiltul min hatzad” its permitted lechatchila all the time, or just in some specific cases?
2.- It is allowed to make “tiltul min hatzad” on basis ledavar ha asur?
Tiltul min hatzad is permitted for the purpose of something that is not muktzeh (letzorech davar hamuttar). It is forbidden if done for the purpose of something that is muktzeh (letzorech davar ha’assur). Therefore, it is permitted to move a mukzteh item min hatzad for the purpose of making space, but not for the purpose of protecting the item itself from the sun.
Sources: See Shulchan Aruch, Orach Chaim 311:8; Mishnah Berurah 311:6, and several places.