You can find
datasheets for the thermocouples (typ. 30mV), thermopile generators (typ.750mV) and the gas valves. The operating current is never mentioned, I faintly remember it could be an amp but losses in resistance are high and any parasitic (metal) junction gives unwanted thermal EMF so the wiring is done one way. Current is something you can't easily measure. The solenoid just has minute hold-in power with mechanical gain (lever) and springs.
Honeywell
V800 standing pilot valves 750mV, 2 ohm coil and 3.4ohms system total resistance gives 220mA.
Honeywell V400 standing pilot valves 30mV, 0.02 ohms system total resistance? gives 1,500mA.
Hot Junction: 1400°F (760°C), Cold Junction: 780°F (416°C)
These systems are rarely used today because they are wasteful the pilot is always on, slow- it can take a long time to react to a flame out and by then, some large burners will build up with fuel which is bad. There are combustion safety standards like NFPA-85, 86 or 87, that dictate the pilot-control response times, which are also dependent on the burner size. Electronic control is the norm nowadays.
For more power look, at thermoelectric generator (TEG) systems. NASA uses Radioisotope Thermoelectric Generators for power on the Mars rovers, Voyager, Apollo etc. i.e.
768 thermocouples for 30VDC. Natural gas is also used for remote areas.