Sorry your experience didn't go so hot. When I got my first Lite-specific Hori screen protector, it went on without a hitch; no evident air bubbles or ghost fog at all. But then when I decided to take it off, it left an incredibly persistent residue that took forever to clean away. (Which is sounding like what you experienced.) From that point on, I went with either the original DS Hori screen protectors that you had linked to, or the English versions of the Lite-specific Hori screen protectors.
It is strange, however that you got such a defective original DS protector - and having scratches and marks already on them is even more peculiar. Are you saying that the inner film was actually scratched up?
As far as the actual applying goes, it can be tricky if you're not sure what you're doing. First, it's imperative to make sure that the screen is completely free from dust and smudges. The other half is the actual application; don't just slap it on, as you're liable to get air bubbles. Take your time and apply it as slowly as possible; I prefer laying it down from left to right, then if I see air bubbles forming, I still have a chance at taking it off and re-trying, before placing it down completely flat. Once it's flat, you would then remove the outer film. If you did it right, you should have little to absolutely zero air bubbles. If you do end up with air bubbles (and they're miniscule enough), you still have a chance at getting rid of them. Try coercing them to the side of the touch screen with your stylus; you can eek them out way.
Can you re-use the screen protector once you've taken it off? I would say yes, assuming you've taken it off with damaging it. But at that point, it's likely that there's smudges and fingerprints to where you were holding it, particularly on the side edges of them, so I don't even bother. They say scotch tape works in getting rid of the smudges and fingerprints, but......
I would give it another try. It's a disaster if you use a shoddy protector, or if you put one on incorrectly. But it works absolute wonders if it's done right.