They're blue stone outcroppings. You can find them on most islands. The example screenshot was taken on Mad Island.

Hit the arrow above the storage button; a menu should pop up. Hit the 'Mode' button and everything will be trimmed down. You probably have something in the way but can't see due to the height of the trees.

I can't speak to the order, but here's the location of all 12 gates.

I totally agree.

I've actually just changed the way I use The Great Oops. IMHO, the greatest value comes from the brains that last 24 hours, but was constantly running out of emeralds needed for stage 3. I found that if you complete stages 1 & 2 (of 4), you only need to use the rope and superglue in stage 3. Leave the emerald and build buttons alone. Hit transform and you'll get 1,000 regular stars, a handful of apparatus stars, collection items, and the 4 brains.

You can cook it or make it in the Louvre.

Do you have another device you can install the game on? If you're Facebook account was linked up, you should be able to get back to your last save point.

Just go through the beginning tutorial, and then connect your Facebook account. It'll ask if you want to use the version that's on your device or in the cloud; if you pick the cloud, you should be able to get back.

I actually do this from time to time, just in case I accidentally spend z-bucks or mess something else up.