I always clean my room in my holidays but my sister always hangs out with her friends. She never stays at home in her holidays. However, my sister and I cook together because we both are interested in food. I play badminton with my friends in the evening, but my sister surfs the internet. She mostly spends her leisure time with those activities to reduce her stress, not like my weekends plan.
Sentence 2: The predicate (main verb) is "hang out", which is third person singular. Who hang out? ="sister" is a noun used as a subject, and it is third person singular. The subject and verb in this sentence agree.
Sentence 4: The first predicate is "my sister and I", which is also third plural. Who cook? = "we" is a pronoun used as a subject, and it is first person plural. Its antecedent is the noun "my sister and I" in sentence 4, which is also first person plural. Therefore, the pronoun and its antecedent agree.