Is it permissible to call a child named "Abdullah" by "Dudu" as a nickname, and is it permissible to mention the name "Abdullah" when the child is in the bathroom?

There is no objection to calling a child "Dudu" as a term of endearment, and there is no offense to Allah's name in that, because "Dudu" or "Aboudi" refers back to the word "Abd" (servant). There is also no objection to calling him "Abdullah" while he is in the bathroom; what is disliked is for a person to mention the name "Abdullah" while in the bathroom, out of reverence for the name of Allah the Almighty.