What do they mean?

[NewHope]

What was hatched a hen must not try to be a rooster

A gift is the key to open the door closed against you

If the dose is nasty, swallow it fast

One hand washes the other, but together they wash the face

[Francois]

Don't try or pretend to be what you're not.
A gift will help breaking the ice, making friends with people
If something is bothersome, do it quickly.
Working as a team lets you do things you cannot do alone.

FRC

[NewHope]

Hi Francois,

Regarding the last one, an American told me it meant:
It takes two hands to wash a face even though it only takes one to wash the other.

It seems opposite to yours.

[Francois]

I dunno. I still prefer my interpretation, but it doesn't mean it is right. Teachers will sort it out.

FRC

[NewHope]

Okay, thank you Francois.
I'd wait to see what is the bottom of this line.