Incorrigibility is proper as defined by the Juvenile Code when: "The juvenile is repeatedly disobedient to the reasonable and lawful commands of his or her parents, guardian, or custodian, and the court finds on the record by clear and convincing evidence that court-accessed services are necessary."
It appears that you cannot file this. The prosecuting attorney can file a petiton as well, but good luck getting them to do that. Unless a crime has been committed, they're going to tell mom to "parent your child."
As to question #2, the answer is no. He can't enter into a contract because he's only 16, so you can't sue him for breaching one. Since he's a minor, his parents/guardians are responsible for his proper care and custody. The apartment complex cannot move against your brother, so they move against your mother instead. They figure if they get rid of mom, they'll also be rid of your brother. To file an eviction, they must be able to cite and prove a breach of the lease and convince a judge that eviction is proper.
If you simply kick your brother out, he'll be homeless. Since mom is responsible for his care/custody, she will then risk an inquiry by Child Protective Services for failing to provide for his support.
Sign up to receive a 3-part series of useful information and advice about child custody law.