It depends a lot on the fine print of the bike, and how good you are at guessing where forces will go. Broadly, if you make a solid structure and mount it properly the bike will usually be fine, but most parts of a bike are not designed for forces in new directions.
Generally muffler tube is 1.6mm or more and mild steel, where even cheap bikes are 0.9mm or thinner and a harder alloy. If you're confident stick welding muffler pipe you can probably weld the thicker parts of a cheap bike, and if you can somehow magic up some really thin rods it'll be much easier. You want to be down around 30 amps for much of the welding, and most transformer stick welders struggle to maintain an arc at that sort of current. Modern switch mode ones work better, even without HF start, because they can up the voltage to keep the arc active and drop it back when the arc starts, on a millisecond-to-millisecond basis. You can't do that by hand :) If you can get gas a TIG kit for a switch mode welder might be worthwhile, but I suspect you might struggle to do that based on your comments.
Normally I wouldn't spit on a gasless MIG, but for you it might be the least worst option if you get the chance to try one in civilisation on day (FFS don't mail order one sight unseen). Maintaining the flux cored wire is not much more hassle than keeping welding rods dry, but you can weld thinner tubing more easily. Or you just just decide to burn 10 or 20 rods getting the hang of thin wall tubing (or just design so you don't need to weld it very much). Running a bead along something solid so you're just barely melting into the thin tube might be the least awful solution. You can't easily join two thin tubes, but you can still do a lot. Or maybe "pre-set your filler" rod by tying non-galvanised wire along the join before you weld, to bulk it up a bit. I haven't done that with stick, but I do it occasionally with TIG with I have something really tricky to weld (a sudden transition from 6mm dropout to 0.6mm tubing, for example)
Welding a new stand plate on, for example, should be fine as long as you can weld the thin wall bicycle tubing. Look closely at the existing plate, most of them are kind of )o( shaped, with the tapered ends designed to give a gentle transition in the amount of force applied, so that the thin tubing of the bike can better cope with the load. Cheaper bikes sometimes fake this with tack welds, or just don't bother because the cheap tubing is thick enough that it doesn't matter. But then you go and load the bike to twice what it's designed for and it does matter.
Panniers and load platforms can often mount to existing attachment points. The hassle is that welding to the seat tube has to be done very carefully, and if you can back purge with TIG that's even better. If you even roughen the inside surface of the seat tube it can make getting the seatpost in difficult to impossible. Filing that smooth requires a half-round file hand patience (or a rotary tool with a filing bit plus skill and luck). You're trying to file 0.1mm off a tube that's 0.9mm thick at most.
Ideally you'll add vertical loads to or very near the axles. If you're lucky the dropouts will be cut rather than cast, and will take welding well. In that case you can just leave enough space to do up the wheelnuts and weld directly to the dropout. Making a rack out of whatever you have, as one piece, is usually fairly easy. Making it light and strong is the hard part/ So give up on light :)
My limited experience is with chopping up bikes to make prototypes, and they seem to survive that better than I expected. Skim the "I built it" section of my site and you'll see a few hacked-about-with bikes. http://moz.geek.nz/mozbike/ and specifically the "Bin Bike prototype" which was a BMX with extras, including a rack largely made of 10mm reinforcing rod.