Is it permissible to purchase mattresses that bear commercial names contrary to Sharia (such as "Sky Mattress" or "Musical Mattress") and images of animate beings, while covering, obliterating, or removing their commercial names, and merely temporarily covering the images of animate beings with bedspreads and sheets?

There is no harm in purchasing products with names that are inappropriate, whether Islamically or customarily. An error in naming does not prohibit the named item itself, nor does it invalidate the sales contract. The ruling pertains to meanings and realities, not to names, and the item being sold must be permissible.

This is supported by the Prophet's (peace be upon him) prohibition of calling grapes "al-Karm" (the noble one), because "al-Karm" refers to the Muslim man. Nevertheless, jurists did not invalidate the sale of grapes if they were called "al-Karm." Rather, the prohibition was for the sake of ادب (proper conduct) and out of dislike.

Therefore, purchasing these products is permissible. The dislike falls upon whoever gave them the disliked name. It is recommended to change an ugly or Islamically impermissible name to a good name when referring to them. As for the small pictures on the mattress, it appears they are in a place of disrespect (imtihan), and their use is not prevented, especially if they are covered.