* type of battery.
* battery specification
* number of batteries
* which battery connection should be made(i.e series or parallel)

1) Many Battery types, Usually the more money you spend, The more efficient, smaller, or lighter.

2)Specs depend on your Loads.

3)Number of Batteries depend on your Loads.

4)Their are Many ways to do Series/Parallel.

What do you want to Backup?

I'd like to sell my small generator (950-1200W) because I'm not sure I can start it, and I'd rather not leave a door or window open enough for the power cord.
I'd like to buy an inverter and battery for use in the house - but what charges the battery?

Second question: Say power is out for a week in 20-30 degree weather. When might copper pipes freeze in unheated rooms if I'm just heating one room with a space heater and a wood stove? None run through outside walls.

I bought an inverter which can be hooked to the vehicle battery and it is kept charged by letting the vehicle idle. Mine is 1500 watts and I have never had to use it. I purchased it for emergency power for my sump pump if the need should arise. Larger inverters are available.

This article says that studies have indicated that pipes burst when temperature surrounding the pipe reach about 20 deg. F. Letting the water run would also prevent freezing.
weather.com - Severe Weather Readiness

That article left out two very important things about draining pipes: toilets and traps, which need to be flushed and then a little plumber's anti freeze poured in each toilet and drain.