The cap itself lets you shrink down to the size of the eponymous Minish. If you say "the Minish" quickly enough, it ends up sounding like "diminish", or to shrink.
The byproduct of the Minish only being seen by children: the most prominent forest-dwelling race is/are the Kokiri/Koroks, who are stated to never grow up; the largest group of Minish live in the woods. No matter how much time passed, they would always be able to communicate.
With the Triforce and Master Sword apparently having been sealed away before the Minish Cap, the Kingdom of Hyrule wasn't able to fend off a wave of monsters and demons that attacked. At least, until the Picori granted The Hero of Men Suspiciously Similar Substitutes in the form of the Light Force and Picori Blade. While they seem to be vastly weaker since they were made by a powerfully magical race as opposed to the Goddesses themselves, perhaps they used the legends of them as a basis for replacements while the real things were sealed away?
Makes sense, and some 100 years later the civil war occurs (the one before OOT) , possibly from people being tired of being left in the dark. Triforces can be seen everywhere, but few seems to know the meaning.
The light force a power that is used to seal away evil and is tied to the royal family is likey just another name for Hyila's power as Zelda should possess the divine blood of the goddess and this isn't the first Zelda's soul was taken away to power up the villain. Demise did the same thing is Skyward Sword. As for the Minish giving the power to them I think they more like unlocked the power since the light force seems to vary similar to the Goddess power.
If you run out of time before the final battle, Vaati obtains the Light Force at the cost of Zelda's life. Since he would then be omnipotent, imagine what he could do. He could have murdered thousands of people, put them all through excrutiating pain, or even destroy all forms of life with little effort. Especially terrifying if you think about the manga adaptation's interpretation, where Vaati stole the cap to become stronger, and was treated badly by Ezlo. Do you think he'd be above getting revenge? The answer is no.