get - become
Grammar question: Can you, dear professional or native speaker, explain me when I can use "get" and "become" interchangeably and WHEN NOT?
Of course, I know: He BECAME a teacher/engeneer/pilot. He gets angry/hungry/ tired. Can he BECOME angry too? I often feel helpless and try to avoid the words in this case. But I cannot imagine to omit them completely.

Question

Answer

With adjectives - yes. The meaning is the same.
With nouns - no.  This changes the meaning.

I became angry = I got angry

I became a teacher =  I obtained teaching qualifications and started working as a teacher ('werden' auf Deutsch)

I got a teacher = I found  a person to give me lessons ('bekommen' auf Deutsch)

Answer

Hi, off the top of my head I can think only of the case you're actually mentioning, that is a change of state. You can "get strong by going to the gym" or you can "become" strong as well, both are equal there. You get old or you become old.  Notice though that in the cotinuous form you tend to "get" rather than become. "I'm getting older and can no longer play football as I used to" "I'm becoming" doesnt sound right. 
Bear in mind also "become" is always more formal. When it comes to emotions though, as you said "become angry", although correct sounds extremely awkward in colloquial contexts. You get angry, you get mad, you get upset, you get sad, you get excited. 
good luck

Answer

good question, sometimes i strugle with this too.

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