-Find another example of a formal system in mathematics, preferably somewhat different from the
examples given in the video. Identify symbols, derivation rules and axioms.-

Although not purely mathematical, phenomenological thermodynamics may in my opinion be considered a formal system. The symbols would then be given by thermodynamic potentials, and the derivation rules would arise from laying out there connections by the four thermodynamic laws (i.e., the axioms).

-Does an axiom have to be “true”? For instance, is it allowed to build a formal system similar to the one discussed in the video, but starting from the axiom “1+1=3”?-

I do not think an axiom must be true, it only requires we should not observe it to be untrue. “1+1=3” could in principle be used to derive a similar formal sytem; the fact that it does not reproduce our N0 additivity may not be known, or even important to us.