a. For sure battery is participating in feeding the load as a 45Ah battery can not be empty in half a day just by powering a 12W katcher...

Hi Jeg. I have doubts about the battery feeding the load directly. I personally think that is probably not the case. 
Menof wrote the following:
"battery need recharge about after day, maybe 7-18 hours working with 700-2000 kw load, then neeed recharge.
In some his video, he showing 4 kilowats how working (fen 2 KW and heater 2 kw)."

It sounds like that with smaller loads the battery can go much of the day without needing recharging.
If the kacher circuit is pulling about 1 A from the battery with smaller loads, then a 45AH battery would
still need to be recharged after about 18 hours if it is not a deep cycle battery, otherwise you could do damage
to the lead acid battery. It seems possible to me that if the kacher is pulling even more than 1 A
when large loads are connected to the output of the device, such as say closer to 2A in that case, then the battery
could need recharging after about 7 or 8 hours.

The thing that makes me wonder about this is if the kacher is drawing closer to 2A under heavy load, I would
think that it would need a bigger heat sink than he seems to be using, as the kacher driver circuit seems to be
not so efficient and the BJT transistor can produce a lot of heat even at 1 A current draw. If the kacher
driver is producing a lot of heat, would he be able to lay a wool hat over the kacher driver for several minutes
without the transistor quickly overheating? He doesn't seem to be using a really large heat sink, although maybe
the heat sink he is using is quite thick. This is something of a puzzle to me though.

His circuit can power loads of two to four kiloWatts. No earth ground wire. Relatively small heat sink on kacher transistor.
Things that make you go hmmm...

When he flings down the device next to the fence then the device is still self-running? He disconnects the loads but never disables the device itself by cutting the back loop. Or did I miss something?

So he doesn't care about short circuiting the wires or sparking, because he knows that device is dead?

If this device is really self-running, can we assume anything is running in a conventional manner. Maybe the components are getting cooler through negative energy! 

Hi Hoppy. I realize you are probably not serious, but anything is possible.
I hope Menof will provide a basic diagram of what this guy explained in his early video from late 2015,
as it sounds like he is possibly doing something unusual with the kacher and tesla coil.
( http://www.youtube.com/watch?v=8GMmIslMhrg )

When he flings down the device next to the fence then the device is still self-running? He disconnects the loads but never disables the device itself by cutting the back loop. Or did I miss something?
So he doesn't care about short circuiting the wires or sparking, because he knows that device is dead?

Hi Zeitmaschine. He has a power switch on the device. He may well have switched off the power switch
on the device before disconnecting the loads.

Hi Hoppy. I realize you are probably not serious, but anything is possible.
I hope Menof will provide a basic diagram of what this guy explained in his early video from late 2015,
as it sounds like he is possibly doing something unusual with the kacher and tesla coil.

Hi Void. Given the very remote possibility that the device is a genuine self-runner, I am serious by saying that the device is not working conventionally! 

Hi Void. Given the very remote possibility that the device is a genuine self-runner, I am serious by saying that the device is not working conventionally! 

Hi Hoppy, Yes, if it is really a self runner, then it would have to be doing something quite out of the ordinary.
That is for certain. With no earth ground on this device, if this is real it would seem it even tops what Kapanadze
has shown, as Kapanadze's device always uses an earth ground when powering several kW's, from what I have seen...

Electric grid also use ground connection and high power genset require grounding too. What's wrong with that ?

Nothing. Could be either that Kapanadze uses.

Electric grid also use ground connection and high power genset require grounding too. What's wrong with that ?

Sorry Forest. Your question doesn't make any sense.
Kapanadze does not use an earth ground connection for safety reasons.
It appears Kapanadze's device needs earth ground to be able to perform well,
if it will even work at all if there is no earth ground connection.