I have had a 5000 watt Sears for years, it runs well pump, furnace, refridgerator, freezer, 1 burner on stove or microwave, and lights, tv (cable usually out when power is anyway).
I have it wired to circuit box with a "main" circuit to switch over from street power to generator, so there will be no power feeding back into the utility lines. Be sure this is done correctly, it is possible to kill someone working on restoring utility service if there is not a switch isolating your house from the outside utilities while the generator is running.
If you do this, you have to be very careful to unplug motors and compressors while starting generator. You also have to be very careful not to overload generator when it is running. Also do not run computer unless you have a good surge protecte/power backup.
You should be in good shape with the 5600 watt, and the envy of the neighborhood when the power is out. You may also find out that you have a lot of new "friends" when they are sitting in the cold and dark.