Is it permissible to wipe over part of the hair and then complete the wiping over the head covering (tarhah) to avoid damaging and drying the hair, and does ablution become invalid by removing the head covering that was wiped over, and is it permissible to wear a jilbab or another head covering and pray with the same ablution?

If the head covering (tarhah) is meant to protect the hair from splitting and is not a khimar (veil), then wiping over it is not permissible, because it is a protective covering and not a khimar. Wiping over a protective covering is impermissible by consensus. However, if the head covering is indeed the khimar that covers the head, then it is hoped that wiping over a part of the head along with completing the wiping over the khimar will suffice. If one wipes over the khimar and then puts on another head covering, the ablution is not invalidated. But if the khimar is removed after wiping over it, then one must repeat the ablution, because the khimar is like the turban in the ruling concerning wiping over it.