No, there's no eviction involved. You just give notice you're terminating the lease. A month-to-month lease can be terminated at any time by either party (you or the tenant) provided the terminating party gives proper notice. If your lease is typical, and your laws are typical, you would need to give this BEFORE the end of September.
If they don't pay the Oct rent, claiming you can "take it out of the security deposit", THEN post a pay or quit and do an eviction based on the unpaid rent. That's an eviction that you can easily win in court.
Again, what does your lease say about termination? What are the local laws? You say you have a lawyer. If you don't know, call and ask them how much notice you must give to terminate a month to month lease.
If you really just want the dog gone and the mess cleaned up, post a notice to remedy those problems or to quit. Maybe that will shake him up enough fix the problems. If nothing else, it creates a paper trail so that if you do eventually want to try to evict him on the basis of the mess, you have some documentation.