A small question for a good start. Is there any difference between
a pressure situation and a pressing situation?
I am looking for a phraze that in the most correct and concise way describes a circumstance of high stress (in formal language).
The phrase you are looking for is "a pressure situation". A "pressing situation" is simply one which requires a quick resolution. It does not necessarily imply that there is any stress involved, albeit that may be the result if the resolution is not forthcoming.
"Do you want soup or salad? Hurry up, the situation is pressing. The waiter needs an answer."
"I resent you always putting me in a pressure situation."