Q1: Is it true that a knot $S^2\hookrightarrow S^4$ has an inverse iff it is trivial? Or it is also an open question?

See relatedly Unknotted $S^{n-2}$ in $S^n$.

Q2: It is easy to see that if a knot $f\colon S^2\hookrightarrow S^4$ has an inverse than its complement $C_f\simeq S^1$. Has the converse been proved?

Both questions are answered below by Daniel Ruberman.