What should we do if we don’t want a roommate living with us and either does the landlord?

If you and the roommate are both renting from the landlord--i.e. the roommate is not subletting from you--you cannot evict him; only the landlord can evict a tenant, and only for proper grounds.

If the landlord wants him out, from what you write, there *may* be grounds for eviction:

1) Nonpayment: as soon as the rent is not paid on time when due, the landlord could bring an eviction action. The problem is, if both you and your roommate are jointly liable for the rent (i.e together you pay the whole amount, and not you each having your own separate payment in the lease), nonpayment of the full amount provides grounds to evict both of you--and conversely, as long as the landlord receives the total amount, he or she cannot evict, even you are paying all of it for your roommate.

(Note, however, that if the roommate does not pay his share and you have to cover it, you would have grounds to sue him to recover his portion.)

2) Habitual late payment of rent, after notice to cease paying late, can also provide grounds to evict, but with the same problem as above--if you and the roommate are jointly responsible, then it's difficult to evict the roommate but not you.

3) If a tenant deliberately or grossly negligently (that is, due to more than ordinary carelessness) damages the landlord's property, that can provide grounds for eviction. It is possible that the cat may provide grounds to evict, if the landlord has warned him about the cat and he had not done anything to stop the damage. It depends in large part on the severity of the damage (de minimis damage does not warrant eviction) and on the roommate have first received notice to cease the damage.