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.
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."
It means to do something surprising. The phrase is associated with 19th Century magicians who performed this trick.
The name of a traditional magician's hat is a top hat. This is the hat that magician's use to pull rabbits, birds and scarves out of.
To do by way of magic.
It was a phrase. "Hold on to your hat, there is going to be a bumpy road ahead"!
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.
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.
Talking through your hat means that you saying something that doesn't necessarily make any sense.