What SSR is it? Regular SS are good only for up to 5 Amp without a heatsink, If you going to use your controller box as heatsink - it should be big enough to dissipate heat generated by 20Amp (around 25Watt)
Yes, it's possible to design your 240v controller box such a way that you can use with 120V only by simple plug adapter. For safety in case of 120V you can install wall outlet with GFCI like this one
Sorry posted a wrong link. Should be US $30.98 |XMT615 Intelligent PID Temperature Controller, Intelligent Temperature Controller, Temperature Controller|Instrument Parts & Accessories| - AliExpress. But anyway it's just an auto translation from Chinese
I have experience with XMT612 and can...
NEMA L14 are three-pole and ground connectors rated for 125/250 V. Intended for three-pole, four-wire hot-hot-neutral-ground circuits .
NEMA L15 are three-pole and ground connectors rated for 250 V. Intended for three-phase circuits.
Are you installing subpanel at your home (L14 )or at some...
Even cheaper - US $2.83 26% OFF|1.37 Inch Programmable Digital Timer Switch Relay Control DC 12V 24V 20A AC 110V 220V 10A Time Controller Delay Relay Module|Timers| - AliExpress
120V 1500Watt is OK for stepping 4 gallons of mash. Though initial heat up and boiling will be slow. So if you can use burner as an aux for preheating and boiling you are fine with 120V . Without a burner it will work as well but will add a couple hours to your brew day.
The "Orange box" is a transformer and , most likely, it's OK. Semiconductors failing much ahead usually. Could you make a picture of area near transformer?