Selfrunning Free Energy devices up to 5 KW from Tariel Kapanadze

[Hoppy]

These are excellent questions.
The only other "perpetual" question is: Where is the energy coming from?
One wire is the ground. The unit does not work without this wire. In addition, one battery wire is still connected during the demonstration.
Is the energy coming through a wire hidden under the battery?
The battery seems to swivel rather than shift when pushed.

Yes, we see at 4:10 that the 'earth' croc clip is removed and at 4:16 he reconnects the battery and the spark gap can be heard functioning. At 4:24 he fails to switch on the light because he has forgotten that the croc clip is disconnected. He then disconnects the battery and reconnects the croc clip at 4:38 after which he is then able to switch on the light. The croc clip is clearly carrying power for the lamp but is not needed to energise the spark gap which is powered from the battery. It appears to me that the other feed wire for the lamp is likely to be hooked into the black battery wire at the battery terminal as questioned by yfree.

[Zeitmaschine]

Is the energy coming through a wire hidden under the battery?
The battery seems to swivel rather than shift when pushed.

Simplest explanation would be a hidden thin wire along of one of the edges of the glass plate. Hence the voltmeter shows 239 volts mains power for a short moment but heavily disturbed due to the EM pollution coming from the spark gap.

But the most interesting TK question would be: How to build a 50Hz resonant LC circuit suitable for 25 amps that fits in TK's tin can using the room left next to a flyback also located in that small tin can. Ideas?

[Hoppy]

But the most interesting TK question would be: How to build a 50Hz resonant LC circuit suitable for 25 amps that fits in TK's tin can using the room left next to a flyback also located in that small tin can. Ideas?

Simples  Not needed! Just run the mains through the tin can to power the lamp via switching and use the rest of the space in the tin for the flyback.

Your thyristors could come into play to chop-up the mains with the gate being switched by a pulsed output from the flyback oscillator. This would account for the silly 25A meter reading. This ensures that the spark gap appears to be creating the generation of 'free energy' to the lamps 

I nearly forgot your Faraday Screen formed by the tin can to possibly reduce EM interference created by the switching. 

[Zeitmaschine]

Simples  Not needed! Just run the mains through the tin can to power the lamp via switching and use the rest of the space in the tin for the flyback.

But this will stress the power meter and the presenters nerves unnecessarily. So what about a resonant LC circuit tuned to the harmonics of 50Hz? The higher the resonant frequency the smaller L and C can be.

[Hoppy]

But this will stress the power meter and the presenters nerves unnecessarily.

Why?

"So what a about a resonant circuit tuned to the harmonics of 50Hz?" Which harmonics?