Does apologizing for not paying the wife's mother constitute an act of disbelief (kufr)? And does the expiation (kaffarah) drop if she forgives it? And how can one expiate for this act and for delaying prayers?

The door of repentance is open, no matter how great the sin. Repentance is achieved by abandoning the sin, feeling remorse, resolving not to return to it, and seeking forgiveness from the person whose rights were violated. If you repented for your mistreatment of your husband's mother, apologized to her, and she accepted your apology, then you have fulfilled your obligation, and rejoice in forgiveness. Delaying the obligatory prayer beyond its prescribed time without a valid excuse is among the major sins, so be diligent in observing it on time, as it is the key to all good.