The issue with the generator is a plugged jet in the bottom of the float bowl. A new carb will fix it or if you are handy, you can remove and clean it. Chemicals won't dissolve those deposits. As the fuel evaporates in storage, it leaves deposit that makes the jet 'smaller'. Be careful not to make the brass jet bigger when you clean it. I've always used a tiny drill bit rotating it VERY carefully with my fingers. These deposits are very hard and brittle and with care can be 'chipped' away with the bit. Sam's Club/Costco is the best source for replacement 'house' batteries. Don't cheap out and use 12 volts. The system works best with 2 6-volts in series. Wiring 12 volts in parallel will cause charging problems since any difference in battery charge between the two results in current flow between them. You really can't troubleshoot problems until the batteries are good. Good luck!
