Is annoying one's mother and her becoming angry due to playing at home considered disobedience to parents (ʿuqūq), and what is the best solution for that?

Causing annoyance or anger to parents is a forbidden act of disobedience. The best solution for playing with your brother is to limit it to what does not disturb your mother. You must respond to her command and obey her with kindness and excellence towards her.