"We live in the Federal Republic of Germany."
Translation:Wir wohnen in der Bundesrepublik Deutschland.
leben is possible; im = in dem not because (die) Republik is a feminine noun and thus the dative form is (der) Republik: in der Bundesrepublik.
I said - Wir leben in der Bundesrepublik Deutschlands and it was marked wrong.
Yes; it's wrong because of the -s on Deutschland.
It's the Federal Republic that is called Germany, not the Federal Republic that is in Germany (like the United States of America).
Similarly, the "City of Hamburg" would be "Stadt Hamburg" and not "Stadt Hamburgs" -- no genitive for this kind of construction.