Successful soldering (you're not welding, and you have the wrong torch if you want to weld) requires that the work be physically clean (scrub it) and chemically clean (flux) - you need a very aggressive flux for stainless, because the reason it IS "stainless" is that it forms a strong oxide layer which protects it from most corrosion. Kester is not going to regard stainless as electrical work, for one thing, so don't bark up that tree too long.
With thin stainless, the other problem you almost certainly have (stainless conducts heat poorly, which makes it very easy to overheat) is overheating the work - once work has been overheated, you have to return to physically cleaning it and refluxing before you have a hope in heck of getting solder on it. You also need to stay out of the "oxidizing cone" of the torch flame - ideally, switch to an oxy-acetlyene torch, make the flame rich, and only use the feather - but you can get there with propane if you're careful. The thickness of the solder does not matter much - you heat the wires, and melt the solder on the wires - don't heat the solder.
You could also move up to brazing, which is a process similar to soldering but at higher temperatures, using different filler metals. When most folks refer to silver soldering, what they actually mean is silver brazing (the rod has some silver, but not a huge amount.) The solder you linked to is a low-temperature solder that has silver in it, from the "lead free" change in "soft solders."
What this is doing on a DIY home improvement site is another question. High voltage electrical projects are not exactly the bread and butter of DIY. Another approach in a lab setting would be to do the whole job in a glovebox full of nitrogen or argon. You could also TIG weld it, if you were good at TIG welding. Spot welding might be another approach, but that will also work better on stainless in an oxygen-free atmosphere (glovebox, etc.)
It's a given that any wires in cordage will be stranded. Being designed for a different kind of service is what makes cordage different from in-wall wiring, and why you should not use one for the other.
So it's a given that any plug whose primary use is cordage is designed to work with stranded wire. Here, you should follow the instructions and labeling on the plugs, because following those instructions are the basis for the testing and certification which the plug received from your NRTL. (Nationally Recognized Testing Laboratory) such as UL, CSA, TUV etc.
Not following the instructions is an "off-label use" (to put it kindly) and can result in an illegal install and non-payout of fire insurance.
Speaking of non-payout, you need to be careful what you are sticking on mains power, and in particular, built into walls. Random electronic components out of the Digi-Key catalog are unfit for use in mains power. Equipment made for use in mains wiring can get its cert/listing faster if it is built with listed components, that is all a component listing means, it does not mean you can use components directly.
So for instance "random relay" is not listed for use in mains, but an "RIB" unit containing that same relay can be used. (RIB ~= Relay In Box).
If your aim is smart appliance control, you may be better off using existing, listed/certified smart devices (i.e. Not Sonoff), and having your electronics communicate with them. Generally, the rules are greatly relaxed for homebrew equipment when it is entirely low voltage and fed by a listed/certified wall wart.
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In this particular case I would use soft yellow sponges. They are cheap and plentiful and easy to replace as you go. They tend to wipe thoroughly.
If they get too corroded/uneven to use (many hours of use) I highly suggest some fine sandpaper and/or steel wool. I'm a little heavy handed but I use a super fine grit on my 1x40" belt sander to reshape and freshen the tips.