What should a Muslim do if exposed to loud noise or obscene language during prayer, and is it permissible for him to use noise-canceling headphones or to interrupt the prayer and delay it?

The basic principle is that a worshiper should not put anything in their ear that prevents them from hearing sounds, unless the sounds are disturbing and affect their khushu' (devotion/humility). In such a case, it is permissible to put something to block the sound.

It is also permissible for the worshiper to make a slight movement to retrieve something to block the sound, but it is not permissible to interrupt a fard (obligatory) prayer except out of necessity.

If it is possible to delay the prayer for a short while without missing its preferred time, then it is permissible. However, delaying it beyond its prescribed time is one of the major sins.