1. This appears to be a PTC resistor, as initially bridge gets voltage thro this and after a while wnen the charging current passes, the ptc gets HOT and increases it value to help reduce inruhs current. meantime the relay operates and tabyepaases the PTC. It is capable of withstanding 3KW for a very short while.
2. U2 is an aux power supply and it is 15V DC as mentioned there. This drives the circuit.
3. YES. all of them are -ve return of the rectified DC of mains supply. (NOT to be grounded and even DON'T TOUCH)
4. same U2 power supply is to be extended to the element on the left also. On right side he connected and left side he showed indication instead of showing a wire.
But beware, as it is NOT sufficient that you know what is shock. the other citizens across globe, espcoially children enthusiastic buit ignorant would suffer fatal shocks. I suppose that such circuits should have the mandatory symbols of skull and bones and a warning .
All the best.