I am not an official expert but I lived through this history and I (think I) remember it;-)
I believe there are three pen lines here. One began with what I think was initially called the Dollar Pen and sold for $1. Sheaffer was not the only company trying to meet this price point. But that was an age of inflation and they could not hold it for long, and then these pens just were described generically as Sheaffer Cartridge Fountain Pens. It was such a cheap pen it did not even come with a model name/number!-) I am not aware of Sheaffer ever calling any of their pens "school pens" in the US of A. In fact, I am not sure if any of the three lines involved here had model names or numbers. The V1-V3 of the OP are three generations of a single line that began with the Dollar Pen.
A second very short line consists of the soft plastic pens with a nib that was hooded in the sense that a bit of plastic protruded over the nib, the protrusion longest at the nib center line. I did not even know about these until fairly recently when one showed up in a lot of miscellaneous fountain pens I bought on eBay. For that one the plastic was pretty well chewed up and the nib seemed inferior to the nibs on any of the other pens discussed here. My guess is that this was an early attempt at a cheap cartridge pen that was not successful and was quickly discontinued.
The third line has nibs that I think are a reference to the inlaid nibs on more expensive Sheaffers, both having a sort of diamond shape. Here the plastic extends out farthest at the edges of the nib rather than the center. I bought one of these new in about 1960, Sheaffer advertised these heavily to high school and college students. The nibs were less flexible than those on the Dollar Pens and came in more widths, tending to the fine side. Most of my haphazardly gathered examples are finer than the nibs marked "F" on the Dollar and its descendants, and at least one is much finer. There is one minor variation that I know of in this line. Some are flat across the top and others are slanted. These sold for $2.95 back then and were certainly intended as a step up from the Dollar Pen. Some people call this model the 295 but I do not recall Sheaffer calling it that.
"Skripsert" was the brand name for the Sheaffer cartridge. I don't think there is a Skripsert pen. The name in a pen ad was just a way of saying it took cartridges rather than bottled ink. (Sheaffer told me back then that they did not make converters because another company held the patent on those, so their pens were at the time either bottle-only or cartridge-only. One consequence of that is that they made no effort to maintain clearance for a yet-non-existent converter, so it is a matter of trial and error fitting converters into these. My experience has been that if you have a bunch of old converters you can always find at least one that fits.)
All these pens used the original Sheaffer cartridges that have a uniform thickness and are flat on both ends. Today cartridges that fit are made in Slovenia and sold under the Sheaffer brand name. A very few Sheaffer pens made after they abandoned Iowa use standard short international cartridges which have a tapered body with a nipple on one end and are flat on the other end. When those came out they were available in many colors but I think it's just black and blue now if you can find them at all. Practically speaking, 99% of cartridges labeled "Sheaffer" will work in these pens, and 0% of cartridges not labeled "Sheaffer" will work as intended. (With a little help from Rube Goldberg you might be able to get 303 Enfield cartridges into these pens;-)
The Made in USA Sheaffer cartridges were identical on their two ends. In fact, Sheaffer advertised this as a feature because most of their competitor's cartridges had to be inserted right way 'round, and it was often not obvious which way was right. The Slovenian cartridges cannot be inserted both ways and work. One end can be punctured by the pen while the other is made to be puncture proof so you can carry an unpunctured cartridge in your pen -- presumably while flying to that important business meeting for which you do not want to show up with a big ink stain on your shirt pocket.
I am doubtful about the Kaweco converters working. They may be slim enough to fit inside the pen, but the Sheaffer cartridges do not have a nipple. I think your best bet would be real Sheaffer converters with metal exteriors that work by squeezing. These tend to be slimmer than the plastic ones that work by twisting.
On the subject of slim pens, there are limits to how thin the pen can be and still take available cartridges. There are some very skinny Chinese pens that use only bottled ink. A stainless Vector might be the most practical solution. Targa is a Sheaffer line, not a Parker line. Sheaffers described as "slim" may take yet another cartridge, this one rare to begin with and long discontinued. If you can find them they are probably dried out by now, but could be refilled with a syringe.
The No Nonsense pens seem to have replaced the two flavors of hooded nib pens as the step up from the Dollar Pen and its descendants. The first generation was made of opaque plastic. A later generation that differed only in using translucent plastic like many Dollar Pens was called the Viewpoint. Both were considerably fatter than the Dollar Pens. These were also available with various calligraphy nibs and in calligraphy sets of various sizes. I think there is a successor today for calligraphy but I don't think any parts interchange with the Made in USA pens.
Plumette's pen is called a Reaktor. It has a crimped nib that I find far inferior to the other pens discussed here. OTOH, I find the cap more attractive than any of the three generations of Dollar Pen caps, and fortunately it is the only part interchangeable with those pens. So I have used these to dress up my old pens for formal gatherings;-)
These cheap Sheaffer pens rather ruined me for more expensive pens, too many of which do not write as well! However, they are no match for the Sheaffer I bought for $25 back when the Dollar Pens cost a dollar.