Is it possible to melt soft metals with household current?
It depends how fast you want to melt soft metals.

Put a screwdriver under a power cable staple behind the wall near the breaker panel and you can even vaporize copper wire and steel in screwdriver  in milliseconds (bang) and spray it all over your plastic safety lens, as I have done in my learning phase 40 yrs ago.(newbie days in Rec Room construction) 

To do it in a controlled fashion only needs some thermodynamic heat equations and sufficient thermal resistance in an insulted ceramic oven and sufficient heat.  

Mass Spectrometers can convert the sample into a plasma with RF energy vacuum, helium/argon burn it and accelerate it past a grid then deflect it and measure the mass of each plasma particle deflected.

A 300W RF coil ( that looks like 1/8" copper plumbing) can make an iron rod red hot in seconds, but not melt it.

A 1000:1 turns ratio spot welder melts metal and welds it instantly by stepping up the AC current in large jaws ( 1 turn output loop) with 10kA at the tungsten electrodes.(preferred)

TIG/MIG/Plasma Welding uses a similar approach with a low current trigger voltage to bridge the gap and then the high current low voltage provides the follow-on energy with I²R [watt] * Rth [degC/W]/time[s] for thermal resistance Rth to get degC/second rise to melting point. where Arc current is inverse to R such that R=k1/I and fuse time = k2 * Amp-seconds [As]