Condescending or supercilious toward
Eat your hat: a statement made when you are positive that something will happen, as in "I'll eat my hat if our team loses this game."
Nothing. The correct idiom is "get OFF your high horse," meaning stop acting so conceited as if you are above everyone else.
It was a phrase. "Hold on to your hat, there is going to be a bumpy road ahead"!
This isn't an idiom - it's talking about some animal with their tail held high, flying behind them.
You had to take your hat off to him based on his success.
Nothing. I'm afraid you've got the idiom incorrectly. Perhaps you actually heard "a bee in her bonnet," which means that someone has an idea that won't go away, as if there were a bee in their hat that they could not ignore.
Literally, this is going to be a fast, rough ride and your hat will blow off if you don't hang on to it. Figuratively, we're going to do something fast with few precautions.
John loved to dance and would do so at the drop of a hat.
This is not an idiom. Idioms make little or no sense unless you know the definition. This sentence makes perfect sense, so it is not an idiom. The dead fish smelled so bad that even as high as Heaven, you could smell them.
The idiom "at the drop of a hat" means to do something immediately or without hesitation, often at a moment's notice or without needing much prompting. It implies a willingness to act quickly or without delay.
The image is of a hat with a lot of names written on slips of paper, and someone randomly picks one. It means that someone randomly picked your name from the available pool of people.
"Sky high" just means very high. You usually hear this as "blown sky high," which would mean either (literally) something exploded and was thrown high in the air, or (figuratively) that someone's plans were thoroughly destroyed.