@Amit C.
From my understanding, a separate checking / saving account will not limit your liability in case your tenant (or anyone else) from suing you for what you've got. Either put the property under its own LLC or bunch up with other properties when you have more, or buy a big enough umbrella insurance to cover those events. Talk to a lawyer to ensure you understand the pros/cons before you form a LLC to do asset protection as you will also have estate planning to think about if you have family to pass assets to.
Talk to your PM to see if they had experience with the home warranty providers and if they recommend the home warranty. Some area don't have good providers so there is no point to waste that money for poor coverage. I have home warranty on some of my DFW rentals to cover my AC. However, I have not replaced my AC on the home warranty but they fixed a few things with $75 deductible each incident. I figure if they replace my AC in 5 years I still come out positive in my warranty spending...