I was not able to find some answers (maybe I am a looser at searching, but still).

When a player is in the brig, he draws skill cards, does not move, then chooses the brig action (that skill check to escape the brig). Skill check succeeds and as a result a player moves somewhere else. Questions: Does he receive one action after his move during the same turn? Must he draw a crisis card?

Is it legal to choose a decision that one cannot fully carry out? For example: A player draws Water Shortage card. It is said, thad you can shoose to loose one food OR to discard three cards and force the president to discard two cards. But either a current player or president does not have enough cards to discard or has no cards at all.

I would be very gratefull for your answers.

A player in the brig does draw skill cards, and cannot move. They may spend their action using the Brig's action (to escape), though they are not obliged to do so. The player receives no further actions, and still must draw a crisis.

As for the second question, I'm pretty sure you can select the option even if you don't have the cards to discard, but I can't find the rule in the FAQ or the rulebook.

In the Card Clarifications section of the rulebook: "Choice Crises: When a player makes a choice on a Crisis Card, he can always choose the top or bottom section of the card, regardless of whether he will be able to fulfill the game text listed (unless the card specifically restricts this)." I believe there is only one Crisis card that restricts you from making that choice, making you choose between sacrificing a nuke or resources.