Minimum of a sum of positive functions is the sum of the minimums of the functions

It is not true. Take $f,g\colon\mathbb R\longrightarrow\mathbb R$ defined by $f(x)=(x-1)^2$ and $g(x)=(x+1)^2$. Then $\min f+\min g=0$, but $\min(f+g)=2$.

$f_1(x)=1$ for all $x$ and $f_2(x)=2$ for all $x$ provides a counterexample.