Is it permissible to mention the name of Allah the Almighty or to include the majestic word [Allah] in speech or action that involves a sin, such as saying "in sha' Allah" (God willing) when planning to commit something forbidden, or uttering it with the intention of not doing something, or using it in the context of singing accompanied by forbidden musical instruments?

It is forbidden to mention the name of Allah when committing sins. The Basmala (saying "Bismillah") may be unlawful when committing forbidden acts such as adultery and drinking alcohol. It is also prohibited to say "Insha'Allah" (Allah willing) when intending to commit a sin, as is dhikr (remembrance of Allah) with songs that contain musical instruments, due to the disrespect this shows to the name of Allah.