Is it permissible to wipe over shoes that do not cover the ankles, while wearing socks that reach the knees, and does ablution remain valid when taking off these shoes for prayer?

It is permissible to wipe over shoes that cover the feet, including the ankles, as they are like khuffs (leather socks). It is not permissible if they do not cover the area of obligation (the feet), and this is the opinion chosen by Ibn Baz and the Permanent Committee. If one removes the shoes after wiping over them while still in a state of purity, their wudu' is not nullified. However, the rukhsah (dispensation) of wiping thereby ends, and if one wears them again and intends to perform wudu', washing the feet becomes obligatory.

As for the three scenarios regarding wiping over socks with short shoes over them that do not cover the ankles:

1. Wiping over the shoes only: Not permissible. 2. Wiping over the socks only: Permissible, and it is permissible to remove the shoes without nullifying the wudu'. 3. Wiping over both the shoes and the socks together: Permissible. If one removes the shoes alone or with the socks, their purity is not nullified, but it is not permissible to wipe over them again in the future until one performs a complete wudu' by washing the feet. The rulings of the khuff apply to socks and covering shoes.