The answer to both your questions is "yes".
a) The surjectivity of $F^*$ already implies that $F:M\to N$ is a closed $C^\infty $ embedding.
This is not very difficult and a proof can be found on page 24 of that book.
b) If $F$ were not surjective, you could take any point $n\in N\setminus F(M)$ outside the image of $F$ and construct a bump function $f\in C^\infty (N)$ with $f(n)=1$ and with support disjoint from the closed submanifold $F(M)$.
Then the equalities $F^*(f)=F^*(0)=0\in C^\infty (N)$ would violate the injectivity of $F^*$.
This contradiction proves that $F$ is in reality surjective and, being also a closed embedding, is in fact a diffeomorphism.