Can my landlord legally do this?

We don't know where you are in the world. We don't know exactly what your lease says.

If you're on a month-to-month lease, the landlord can change terms at any time. If you don't agree to them, it's essentially notice of termination.

Even if your lease says small animals are allowed, it may be with landlord approval. You also may have misunderstood what was meant. For example, they may mean gerbils in cages, not cats roaming around the property. It's good to err on the side of confirming with them that something is okay, rather than getting something and upsetting them.

Even if you were on an annual lease before, if you didn't renew as an annual lease, you're probably month to month now. If the landlord now is saying no pets, if you don't like it, you need to move.

If you want better answers, we need to know more background information like what type of lease you're on at this stage, and word for word what your lease says.

Depends on the lease. If it says small animals are allowed with permission, then maybe. It also depends on the definition of small animal (some consider that to be fish or small caged animals like hamsters.)

If you are on a 30 day rental rather than a lease, if you don't get rid of the cat, then they can ask you to leave.

no, landlord seems irrational and is violating the lease.

Yes, if you don't like it move, his house his rules.

Yes it is.