Is he permitted to abandon his mother, who married عرفيًا (informally) and then denied it before admitting it, and who has suspicious relationships, knowing that she raised him and spent on him?

Your mother did not lie. An "urfi" marriage (customary marriage) is valid, even without a guardian, according to some jurists. It is also valid if the intention was not to officially register it. Repentance is obligatory, and you must seek her forgiveness.

Do not think of killing her. You must advise her kindly, even if she has committed a sin. If she insists on committing a forbidden act, she can be confined or restrained to prevent her from immorality.

What you did, insulting the man's mother, is forbidden. It is one of the gravest sins for a man to insult his parents.