Ungalvanized (you only get galvanized if you ask for it that way) will be cheaper, but be warned: when they manufacture steel thay coat it with an oil to keep the rust down. That crap gets everywhere. And you will be amazed at how fast it rusts after you wipe off the oil, especially your welds.
You will have to prime & paint this, which is not easy. You should spray it, and you really shouldn't use spray cans. It's like painting a car, you really have to know what you are doing for it to come out nice.
I think 1/16" is too thin, I would go with 1/8" (20 gauge). I don't think it would make it any more heavier than a cabinet built froom wood. Since I haven't seen your plans I can't estimate what it would weigh.
Ask around, I can readily buy sheet steel in 4x8 sheets (just like plywood)!! Steel supppliers are the best. There is a steel mill about 40 miles from where I live, but they only sell by the truckload.
If you are going to flame cut, be prepared to spend LOTS of time with your grinder. You just won't get a nice edge that way. A plasma cutter would work much better, but those are really expensive.
Zero margin for errors? What do you mean by that? Anyone with welding experience can see that there is plenty of room for error with pieces that are not cut exactly right. Cut a little more off, or fill the gap with weld. It is just like woodworking.
The whole point of the joints is that they are NOT flexible. This is what gives your structure the strength it needs. The more jounts you can eliminate by using bent sheet steel, the better. A welded joint is stronger than the steel it connects.
Having said all this, I don't even want to think about all the problems you will have with connecting your controls, computer, etc. so they don't ground out on your cabinet. I doubt that there will be any effect on your monitor being in a metal cabinet. Anyway, I would love to see you do this, I think it could be a really awesome project. You've got your work cut out for you here.