Is it permissible to perform istinja' with the left hand on which the word "Allah" sometimes appears, whether visible or hidden? And what is the Shar'i ruling on that?

The legal principle is that everything that entails honor and respect is preferably done with the right hand, and its opposite with the left; due to the dignity and honor of the right hand. As for what your friend mentioned, we do not see any legal basis for it to be relied upon.