Can the landlord just change her mind after our verbal agreement regarding my dog?

If you relied on verbal agreement and you have honored the terms you can fight any action she might take on those grounds. You should consult a real estate attorney to review all of facts and assist you.

Yes, because you did not get it in writing. I suppose you might make a claim that the new carpet was consideration for the dog, but without it being in writing, she can more than likely make you stick to the written agreement.

All such lease arrangement should always be in writing for reasons that are obvious now. This does not mean that oral promises can be broken at will. Certainly, the receipt for the new floors would be evidence that something out of the ordinary is going on. If she attempts eviction based on a breach of the lease, you would seem to have a good defense so long as you can persuade the court that your version of events is the truth.

I would fight it in court. Send her a letter recounting the history via certified mail, return receipt requested. Then, if she tries to evict you, show the Judge the letter and tell your story. Bring any witnesses.