When no one was wearing the invisibility cloak, how could they find it? Wouldn't it be invisible? I read through this and it explained its history, strengths and weaknesses, but didn't really answer my question.
The cloak is visible from one side. If you look at the scene where Harry gets it for a present, you can see it from the "inside" side.
While there are no actual clips from the movie on YouTube, here is the scene done in Rifftrax.
You can also see it in the picture in the Harry Potter wiki article on the invisibility cloak (which you linked to). Granted, the wiki doesn't address it, and at first glance at that first image of it, the design is kind of funky (Rifftrax describes it as "tie-dye velvet") so it's hard to be sure you're seeing the inside of the cloak, but if you watch the scene in the movie, it's clear the cloak is visible from the inside.
From the first time Harry (and we) were introduced to the cloak
This only left one parcel. Harry picked it up and felt it. It was very light. He unwrapped it. Something fluid and silvery gray went slithering to the floor where it lay in gleaming folds. Ron gasped. "I've heard of those," he said in a hushed voice, dropping the box of Every Flavor Beans he'd gotten from Hermione. "If that's what I think it is -- they're really rare, and really valuable." "What is it?" Harry picked the shining, silvery cloth off the floor. It was strange to the touch, like water woven into material. "It's an invisibility cloak," said Ron, a look of awe on his face. "I'm sure it is -- try it on."
JKR keeps up this description all through the books. So you can definitely see it (it also seems to feel differently than normal material).
I thought that the cloak was just a cloak until it was put over someone and it then, for want of a better phrase, engaged it's cloaking device.
Otherwise how could it be made (how do you stitch something that cannot be seen? When you come back in the morning, how do you know where you got to?)?
What if you dropped it?
I am not sure whether this is mentioned tbh but it is the only thing that makes sense to me. Have the cloak visable from one side and not the other does not sit right in my mind, what if it fell invisible side up? What if the cloak was twisted and the invisible side up was caught under something, you would rip it when you tried to pick it up.
One side of the cloak was not visible.
Borrowed from http://harrypotter.wikia.com/wiki/Cloak_of_Invisibility