A fall into a pit, a gain in your wit

I like the first one, yet I have no clue whether they actually do work in English speaking countries - me just yet another alien to the English world.

I have heard "Once bitten twice shy" used, but the others do not seem to have been adopted into English 'sayings'.

But they are not really 'idioms'; an idiom is a 'way of saying something' that does not make sense without an 'inside' familiarity with the phrase, like :
"horse of a different colour" - meaning something very different from the idea that was being discussed.

The examples you gave are called 'aphorisms' or 'sayings'; 'village' wisdom.

Yeah, just like:
Hadn't the bunny rested for having a $hit, the fox wouldn't have eaten it (the bunny).
(translated from German language)

Yes BoGoWo, they are aphorisms.

"A man who carries a cat by the tail learns something he can learn in no other way."
- Mark Twain