Show that the fibre $\pi^{-1}(p)$ is a regular submanifold Exercise 6.4 of the book Manifolds and Differential Geometry (Jeffrey M. Lee, 2009) states the following:
Show that if $(E,M,\pi, F)$ is a (smooth) fiber bundle, then
$\pi: E \rightarrow M$ is a submersion and each fiber $\pi^{-1}(p)$ is a regular submanifold
which is diffeomorphic to $F$. Show that if both $F$ and $M$ are connected,
then E is connected.
I think I can proof the first statement: Because $(E,M,\pi,F)$ is a fibre bundle, for each $p\in M$ there exists a neighborhood $U$ which contains $p$ and a diffeomorphism $\Psi:\pi^{-1}(U) \rightarrow U\times F$ such that $\pi|_U=\Psi\circ \mbox{proj}_U$ (composition is by the right). Hence $d\pi|_U = d\Psi\circ d\,\mbox{proj}_U$; since $\Psi$ is a diffeomorphism, $d\Psi$ is bijective. Clearly $d\,\mbox{proj}_U$ has constant rank equal to $\mbox{dim }(M)$ and thus it follows that $\pi$ is a submersion.
In order to see that $\pi^{-1}(p)$ for $p\in M$ is a regular submanifold of $E$ of dimension $k$, I have to find a chart $(A,\tau)$ of the smooth manifold $E$ such that
$$ \tau\big(A\cap\pi^{-1}(p)\big) = \tau(A)\cap)\big(\mathbb{R}^k \times\{c\}\big), $$
where $c$ is an element of $\mathbb{R}^{d-k}$ (typically $c=0$) and $d=\mbox{dim }(E)$.
So, my questions are:
1- Is right my proof? 
2.- How I can see the fibre over $p$ is a regular submanifold? How proof that is diffeomorphic to $F$?
3.- If $M$ and $F$ are connected, then $U\times F$ is connected for all open set $U\subset M$ and thus the open set $\pi^{-1}(U)$ is connected in $E$ thanks to the diffeomorphism $\Psi$. So it is clear that $E$ is locally connected. However, the connectedness of $E$ has not proof yet. How is the proof?

Just to give an additional reference: In the book Manifolds, Groups, Bundles, and Spacetime by R. Dale Gray, there is a proof that $\pi$ is a submersion (Theorem $8.1.13$) and that the fibers are regular submanifolds diffeomorphic to $F$ (Theorem $8.1.14$).
