Do not send the money. Do not rent without a written agreement. And in light of the increase in landlord-tenant abuse I've seen in the last couple years, I wouldn't sign the lease without letting a lawyer review it and explain it to you. ssAnd yes, an out of state landlord should have a local rep: who do you go to when the AC goes out?
Agreed. Any claims against the HOA grow out of the CC&Rs, and I doubt seriously you're going to find a basis for suing the HOA there. But there is clearly a claim against the neighbor. And rather than pay the HOA, tell them (in writing) who is doing the damage and direct them to send their bill to the neighbor.
If you are paid up on the rent, I think your defense should work. However, don't expect the Justice Court judge to be familiar with this statute. Be prepared to explain slowly and in detail. A particularly good fact is that the landlord renewed the lease after knowledge of the dog. The key fact will be proving that you got the dog in August and that you have lived next door to the landlord since that time (thus, indirectly establishing that the landlord knew of the dog, even if he denies it).