Ok, the second example where capacitor is connected directly to a battery with no resistor. Do this statements hold true ?
1) The first time we connect a capacitor to a battery we will get a spark and stress both battery and capacitor
2) Everytime the spot welder turns on, both capacitor and battery will supply as much as they can to the spot welder
3) When that happens, the capacitor will get empty and in effect again stress the battery because it will charge it super fast with no resistor
1) likely to be true, but I would say that the battery is likely to suffer more than the capacitor.
2) sort of true
3) not true
It's all a bit complicated, because the amount of power that the spot welder takes will depend on how much the voltage dips and for how long it dips when the weld happens. The amount that the voltage dips depends on the characteristics of the battery and the capacitor.
A capacitor that is connected to a circuit will reduce the amount that the voltage dips, so that that will reduce the peak current taken from the battery, and make the welder produce more power.
The capacitor will not become completely discharged, because the voltage will stay above zero, so the charging up of the capacitor after the weld will be less severe than connecting a completely discharged capacitor.
With a capacitor connected, the overall charge taken from the battery will probably increase, because the welder most likely takes more power if the voltage dips less and the capacitor has to be charged back up after the weld.
There are a lot of factors that come into this. An important one is how the welder behaves when the voltage dips. It may take less power (and make a worse weld) or it could take more power and stress the battery more, or something in between.
Another important factor is the internal resistance of the battery, so how much it dips for a certain current. The actual capacity of the battery isn't so important because the weld and the capacitor charging is too short to take a significant fraction of the battery capacity.
The capacitance and internal resistance of the capacitor is important. The internal resistance will control how much it helps the voltage dip, and the capacitance will define how long it can last for.
I was recently looking at a 12 V system where a battery was supplied from a power supply, and a motor would take about 500 A when starting. The battery and the power supply each had internal resistances of about 20 mOhm, so the voltage would dip by about 5 V, with about 250 A coming from the power supply and 250 A coming from the battery.
Of course, the voltage drop did reduce the peak current to the motor.
A supercapacitor of 500 F and about 5 mOhm would take a lot of the peak current, because it's resistance was lower than the battery or the power supply. That reduced the voltage dip and the currents taken from the battery and the power supply. At 500 F, the capacitor voltage would reduce at a rate of about 1 V/s when the motor ran, but the start-up was only 20 ms or so, and that didn't matter.