travelingpac: your are talking about two different insurances here because the German system is a bit different:
1) there is an insurance which covers all the contents, valuables and property in your rented place against theft, vandalism or destruction by water, fire and so on. This is often called a renter's insurance or tenant content insurance. This insurance is called in German HAUSRAT Versicherung. It does NOT cover damage that you, a family member or the dogs would cause to another person's property or bodily injury. The premium for this insurance is depending on size of rented place in squaremeter, area/region of the place and total of property valuable. Usually the value is computed by insurances as a rule-of-the-thumb at 650.- EUr per squaremeter.
2) the insurance covering damages caused by you, family members or dogs to other people's property or bodily injury is a third party liability insurance or in GErman PRIVATE HAFTPFLICHT VERSICHERUNG. This is actually next to good
health insurance the most important insurance for anyone. For you and the family the costs will be around 80-120 EUR/year, the dogs will come extra and the premium will be depending on size and race of dogs.
the landlord insures the building against fire and damages. you don't have to take care of that (but you will be charged a share of these costs through your "Hausgeld" or "Wohngeld" indirectly.
you should take the services of an independent broker in order to find the best solution for you. Otherwise and if your German is up to it, try to check the independent consumer protection magazin FINANZTEST in order to find an insurance where you get your money's worth of protection and coverage and which has a good track record for claim settlment.
Cheerio and good luck,