I think you heard the idiom wrong. It should be "by the skin of her teeth," meaning that she barely succeeded.
This is not an idiom - when you compare two things with "Like" or "As" then you have a simile. This comparison says that something is as hard as pulling teeth out would be. You often hear this said when someone is very reluctant to part with information and it must be pried out of them like pulling teeth one by one.
An idiom is a rendition of a combination of words that have a figurative meaning. Most idioms have no clear "inventor".
In brief it seems falsely translated from the common idiom 'to pull someone's leg', which has the meaning "to trick/fool/kid someone". For example: "Are you pulling my leg?" (Br.) / "Are you kidding me?" (Am.)
It means its hard to pull someone teeth without them screaming and yelling or whatever which means it hard to teach you something without you not getting it or asking a lot of difficult question.
This is not an idiom, so it means what it says. Something is as rare or hard to find as teeth on a female chicken (or any chicken) - chickens are birds, and do not have teeth.
Pulling Teeth - band - was created in 2005.
"To be" is not an idiom - it's a verb.
No, because you have used the term incorrectly. The idiom is YOU'RE pulling my leg, as in YOU ARE doing it."I know you're just pulling my leg when you say you can fly."
Yanking your chain,' means to insult someone in a teasing manner, perhaps convincing them of something untrue. This is also called 'pulling your leg.'
Pulling the wool over one's eyes means fooling them -- they are saying "don't try to fool me."
It means you are lying to me or trying to fool me.