Yes, you can use a regular battery charger on your batteries. Since they are connected in parallel, make sure the charger connections are to the + of one battery and the - of the other. Take care not to reverse the polarity of the connections.
Charging 10 min is a common error to those who are unfamiliar with batteries and electricity in general. Just starting the engine will take more amperage that the gen set and or alternator can put back into the batteries....so each time you did that, you depleted the batteries even more than when you started.
I agree with others...your first concern should be that electrical problem. Remember...electricity can KILL! It doesn't take a whole lot of amperage to do some serious damage to you or others.
The second thing you should do is go down to Walmart or Harbor Freight an buy a cheap multimeter and use it to measure actual voltage on the batteries and the outlet where you are trying to plug your rig into.
For the batteries, using the DC Voltage function on the multimeter, 12.6 is full charge whereas 12.2 is only 50% charged. When running the gen set or alternator, you should be reading well over 13v, maybe even 14v on the batteries.
For the house outlet that you are trying to plug into, set your multimeter on AC Voltage, and measure the voltage on the two slotted holes in the outlet. You should read 110 to 120 vac. Then put the red lead into the small slotted hole and the black lead in the round hole. Again you should read 110 to 120 vac. Then stick the red lead in the large slotted hole and the black lead in the round hole...you should read 0 volts. Anything other than these readings indicates a faulty or mis-wired outlet.
You may have to remove the cover off of the outlet to see if there is a green wire attached to the very bottom of the outlet. This will be the ground connection and it sounds like your outlet may not be grounded. Be careful with the cover off and don't touch anything on the outlet until you have the cover back on.
Do some research (use the search engine here on rv.net) and educate yourself.
Good luck and I hope this helps
Ron