Is it permissible to remain silent in the face of mockery towards Allah Almighty and to suffice with only an inward disapproval? What is the ruling on someone who utters such words and insists upon them? What is the opinion regarding expelling the one who utters them from the home?

This statement is disbelief in Allah Almighty because it contains an element of belittling Him. Its utterer must repent and renew his faith, otherwise, he is a disbelieving apostate. Your denunciation of evil is an obligation, and this statement is among the greatest evils. Your companions should have denounced it. As for playing chess, it is forbidden by scholarly consensus if it distracts from an obligation or involves something unlawful. There is a difference of opinion if it does not distract and is not accompanied by anything unlawful, but the majority of jurists consider it forbidden, which is what the Companions ruled. It is obligatory to abandon this game and repent from it.