Yes. A similar expression is "A bird in hand is worth two in the bush". It means: don't assume something is accomplished or acquired before it's actually done.

Sure. It's still used. Since I have chickens I've learned that there are a great many expressions that have to do with chickens that used frequently. :D to be henpecked (to be nagged) to fly the coop (to leave home) to be a mother hen (to be over-protective) to lay an egg (to fail) There are a great many more.

Yes, it is. But often we just say the first part: 'Don't count your chickens'. This is the case for all such sayings.

Hello Carlos: Yes, the old adage is often used.