Let $M$ and $N$ be smooth manifolds and $f: M \to N$ a smooth map. Define the pullback bundle $\pi_f^*:f^*(T^*N) \to M$ as usual by $ f^*(T^*N) = \{(x,j^1_{f(x)}g) \in M \times T^*N \}$ with projection $\pi^*_f(x,j^1_{f(x)}g)=x$ then the pullback of $f$ is a smooth morphism $f^*: f^*(T^*N) \to T^*M$ and moreover a vector bundle morphism over the identity defined by $f^*(x,j^1_{f(x)}g) = j^1_{x}(g \circ f)$. Now the questions are:
1.) Suppose $f$ is an embedding. Is $f^*$ a surjective submersion?
2.) Suppose $f$ is a surjective submersion. Is $f^*$ an embedding?
3.) Suppose $f$ is a diffeomorphism. Is $f^*$ a diffeomorphism?
(Regarded $f^*$ just a smooth map, forgetting the additional bundle structure)
4.) Suppose $f$ is a diffeomorphism. Is $f^*$ a vector bundle isomorphism?