As a landlord you need to do yourself the favor of knowing the law. There's a pretty good summary available here - I recommend reading it, cover to cover.

To start with, consider getting credit checks on all prospective tenants including past histories of eviction. There are some tenants who have been through the process enough times that they know more about eviction law than most lawyers. I have seen them drag out eviction cases for months, while the landlord collects no rent, sometimes topped with the ultimate kicker - you win and just as you're about to get them out they file for bankruptcy protection and get even more free time in the apartment while you either spend money trying to lift the automatic stay or wait for it to expire.

In terms of heat, the first question is whether or not there's a problem with the heating in the rental unit. It needs to be in proper working order. Beyond that, specific minimum temperatures for rental housing can be set by ordinance. For example, San Francisco's Housing Code requires a permanent heating source (not a space heater) that maintains a temperature of 68 degrees between 5 am and 11 am and between 3 pm and 10 pm.