I am assuming you are going to be using a photo emulsion, is this correct?
The key to creating artwork for photo emulsion you need an opaque image on clear film - this is known as a film positive. Of the 3 pieces you linked to, the only one that will be "easy" is the jinalotus.gif, as it is a solid image with no gradations, and is a single color. This is a good beginners starting place. To make this into a film positive you can output it on a good laser printer at the size you want to print. Personally, I prefer to use commercial film positives output on equipment that is made for the purpose - any good repro graphics house can do that for you... but if you don't have easy access to a repro house, then just try outputing it yourself on a laser printer. To make it opaque enough, you might want to make two copies and double them up when you shoot your screen. The best film positives are black, although red can work well too.
The other images (the fish and the lotus) both have gradations (where colors blend from one intensity to another, or from one color to another) - this requires very advanced techniques both in separating out the colors as well as shooting the screens and doing the printing - which requires halftone film positives. Anyway - for the time being you should stick with the simple, one color images.
You can print on tights, and by using the proper inks you should be OK - however, the image will likely stretch so you might want to experiment. And never try to print on tights while someone is wearing them - trying to get them through a dryer is a real hassle.
As far as saving a stencil - you save the frame. Trying to remove a stencil and then restretch it is not worth the bother - it will almost never work out well. Many shirt printing shops just save the screens for work that repeats often.
It will depend on the nature of the image. However, if you are just doing one it would probably be better to work with a common stencil technique rather than screen printing - particularly if you want to do a number of colors or blends and things like that.
Screen printing is great for doing a number of the same thing, but it takes a lot of experience to get a good result, costs a great deal more than common stenciling, and is a lot of work if you are just doing one piece.
I have found that there is little you can do that will give you good results as far as removing the printed areas. Perhaps someone will have a suggestion - but a different approach would be to print over the existing print with a dark color panel. Of course, this would require that it somehow fits into the new design you want to put on the shirt so it might not be a useful solution.
I recall once, over 25 years ago, where a friend of mine had printed 2000 shirts for the USS Cleveland (which is a Navy ship) but accidentally printed "USS Clevland" - leaving out the second "e". We used those shirts for rags, running shirts, sleeping shirts, and you-name-it for a long time. They were very nice shirts and lasted a long time. He had boxes of them for several years. A nice reminder that you need to get the spelling right most of the time. Of course, the point is, if he could have somehow cleaned the ink out of the shirts after they were printed and heat-set it might have saved the day - but even if he could have removed the ink, the labor of doing so would have been a great deal more than just buying new shirts, which, of course, what he had to do.
There are ways to blast the ink out of a shirt that has been printed but not yet gone through the drying tunnel - but that is really only usefull on the occasional small smudge and things like that. It just isn't effective on anything big.
Actually, screenprinting is a type of stenciling, but very different than common stenciling.
That is true, but while you could say screen printing is stencilling, you can't say that stencilling is screen printing
Exactly. You can say that a dog is an animal, but not that an animal is a dog. And even more so, neither dogs nor animals are screen printing.
It might be worthwhile to note that screen printing does not necessarily mean using photo emulsion. There are numerous alternative ways to make a screen printing stencil, including cut or torn paper, glue & touche, hand cut or photo exposed water or lacquer adhering films, reductive hand applied blockout, and so on.