The main battery is 400V and the charger in the car needs at least 10 amps at 110V to operate. In principle you could do this from a 12V battery - there are "inverters" which will generate 110V for you. But the current at 12V would be around 100amps, so the battery will have to be quite hefty. If you have a solar powered home installation with a good storage battery (such as one from Tesla) then you may be able to do it.
Your idea of a "reversible" charger is one that is being considered by some car manufacturers - it has been realised the most electric cars sit idle and the battery could be used as a reservoir of green energy from solar or wind. It is a burgeoning field, and one that merits monitoring. A simple idea would be to store mains power at off-peak times and then return it to the grid at high peak times. But this is all in the future unless you have the resources to start tinkering.
Let me know if you want some references to the developments.