I think the following argument answers your question. It reminds me a lot of the exercises in the basic topology course I took during my undergrad. Fun times.
Recall that $\pi$ is a submersion, so for any $x$ in $E$ there exists a neighborhood of the form $U \times V$, such that $\pi$ identifies with the projection $U \times V \to V$. Suppose we only know that $E_t := \pi^{-1}(t)$ is compact for any $t$ in $B$, and let's show that $\pi$ is proper.
Let $K \subset B$ be compact and set $L = \pi^{-1}(K)$. Let $(\mathcal U_\alpha)$ be an covering of $L$ by open subsets of $E$. We will show that there exists a finite covering of $E$ by open subsets of $(\mathcal U_\alpha)$, and thus by sets of the original covering.
Take $t \in K$. Find neighborhoods $\mathcal U_1, \ldots \mathcal U_{n(t)}$ that cover $E_t$. For any point $x \in E_t$, find neighborhoods $U_{t,x} \times V_{t,x} \subset \mathcal U_{j(x)}$ for some $j(x)$ (the $\mathcal U_{j(x)}$ being one of the above), such that the map $\pi$ identifies with the projection $U_{t,x} \times V_{t,x} \to V_{t,x}$.
The compactness of $E_t$ gives finitely many such $U_{t,x_\nu} \times V_{t,x_\nu}$ which cover $E_t$ and such that there is a $V_t \subset B$ contained in the image of any $V_{t,x_\nu}$ by $\pi$ (the intersection of the finitely many such $V$ contains $t$).
Now: $K$ is compact, so it is covered by finitely many of those $V_t$. The corresponding finite collection $(U_{t,x_\nu} \times V_{t,x_\nu})$ covers $L$, and consists of subsets of elements of $(\mathcal U_\alpha)$. Thus finitely many elements of $(\mathcal U_\alpha)$ cover $L$.
For the first question, if the dimension of $E$ is greater than or equal to the dimension of $M$, then having maximal rank is the same as the differential map being surjective. The other direction is also obvious.
Since we are dealing with differentiable manifolds and we have that property on the differential map, the manifold $E$ has a local product structure, i.e. for any $e\in E$ there is a neighborhood $U$ of $e$ and a manifold $F_e$ such that $U$ is diffeomorphic to $\pi(U)\times F_e$ (this is a direct consequence of the implicit function theorem). However, this product structure is not restrictive enough and you might end up, for example, with fibres with different topologies or homotopy types, for example the fibred manifold $(\mathbb{R}^2-\{0\},pr,\mathbb{R})$ where $pr:\mathbb{R}^2-\{0\}\rightarrow\mathbb{R}$ is the projection to the first coordinate, illustrates that phenomenon (by the way, this is also an example of a non-proper map which is not a fiber bundle). So, with only that product structure is hard to relate the topology of the manifolds involved. However, once you impose the local trivialization condition, you get rid of those "pathologies" (for instance the fibres on each component of the base become diffeomorphic). Also, local trivializations give the fibre bundle the homotopy lifting property, so you can compute invariants of the manifolds via exact sequences of homotopy groups, or the Serre spectral sequence, etc.
I'm sure that local trivializations do much more for us, but that's what I know so far.
Another fibred manifold which is not a fiber bundle is the following (non-proper) map: define $\mathbb{R}^2/\mathbb{Z}^2\rightarrow\mathbb{R}$ by $(x,y)$ mod $\mathbb{Z}^2\mapsto y-\sqrt{2}x$. Then this is a surjective sumbersion onto its image which is not locally trivial.
Best Answer
Let $p\in N$, there exists a neighborhood $U$ and a diffeomorphism $\Psi:F^{-1}(U)\rightarrow U\times F^{-1}(p)$ such that : $F|_{F^{-1}(U)}=\pi\circ\Psi$. First, let's consider $q\in U$. Define $\pi':U\times F^{-1}(p)\rightarrow F^{-1}(p)$ the other projection. And define also $\Theta:=\pi'\circ\Psi:F^{-1}(q)\rightarrow F^{-1}(p)$. $\Theta$ is a bijection whose inverse is given explicitely by $\Theta^{-1}:F^{-1}(p)\rightarrow F^{-1}(q),y\mapsto\Psi^{-1}(q,y)$. It is not hard to see that $\Theta$ and $\Theta^{-1}$ are both smooth maps.
Now let's consider a point $z\notin U$, so there exists a neighborhood $V$ and a diffeomorphism $\Psi':F^{-1}(V)\rightarrow V\times F^{-1}(z)$ such that : $F|_{F^{-1}(V)}=\pi\circ\Psi$. If suppose that there exists a point $q\in U\cap V$, then there exist two diffeomorphisms $\Theta_1:F^{-1}(p)\rightarrow F^{-1}(q)$ and $\Theta_2:F^{-1}(q)\rightarrow F^{-1}(z)$. $\Theta_2\circ\Theta_1$ does the job.
Let's suppose $N$ is connected, so there's a path from $p$ to $z$. We can cover this path with a finite set of open balls that interset one after the other. So the final result is proven.