1. Why doesn't the ring turn Sauron invisible like it does with everyone else who wears it?
The invisibility is an effect of being subjugated to Sauron's will and being transformed into a wraith in his service (hence it causes you to enter wraith-world). It's never stated explicitly in the film, but if the Ringwraiths did not wear their black cloaks they would be invisible, having been made permanent servants of Sauron and thus appearing only in wraith world.
Sauron being owner of the Ring, and it being his power infusing it, he is obviously not dominated by the Ring and therefore not forced wholly into wraith-world when he wears it.
That of course is the retconned answer. The real answer is that Tolkien's first notion was an invisibility ring and he was stuck with invisibility as a side effect of the Ring when its significance was further developed.
2. Are the big badass creatures like the Watcher In The Water, the Balrog and Shelob one of a kind monsters or are they part of a larger species?
The Watcher, somewhat uncharacteristically for Tolkien, was never developed or explained. The Balrog is a Maiar spirit like Gandalf which chose to take the form of a demon rather than an old man. Shelob is "the last child of Ungoliant," who was an evil spirit in the form of a spider.
If they are part of larger species, where the fuck are the rest of them during this story?
In the larger world outside the story. If you think what's seen in this story represents everything Middle-Earth has to offer, you could not possibly miss the point more.