Han er blevet berømt for sine sange i 1970’erne.

Your sentence gives the impression that he has lately become famous for songs he made in the 70s.
I assume that what you mean is that he became famous in the 70s and continues to be so today, so I have made this correction.

Han blev indkaldt tilaf hæren til at kæmpe i Afrika, men Fausto kæmpede ikke der.

There are (at least) two ways to formulate this sentece. One is the one I've written, the other is "Han blev indkaldt til hæren for at kæmpe i Afrika, men Fausto kæmpede ikke."

I have deleted "der", because it implies that he didn't fight in Africa, but fought somewhere else. If it was the case that he refused to fight, you could write "..., men Fausto nægtede at kæmpe."