Here is a counter-example to Q1. Consider a Reeb-like foliation on the annulus $E=S^1\times\mathbb R$. Namely fix a point $p\in S^1$, let $I=S^1\setminus \{p\}$ and fix a smooth function $h:I\to\mathbb R$ which tends to $+\infty$ at both ends of the interval $I$. One leaf of the foliation is $\{p\}\times\mathbb R$, and the remaining strip $I\times\mathbb R$ is foliated by graphs of the form $y=h(x)+const$. Note that all leaves are diffeomorphic to $\mathbb R$.

There is a map $f:E\to S^1$ whose fibers are leaves of this foliation. First define $g:I\times\mathbb R\to(-\pi/2,\pi/2)$ by $g(x,y)=\arctan(y-f(x))$. Extend $g$ to the entire annulus $S^1\times\mathbb R$ by setting $g(p,y)=-\pi/2$. Now we have a continuous map $g$ from the annulus onto $[-\pi/2,\pi/2)$ whose fibers are leaves of our foliation. It remains to compose it with a continuous bijection from $[-\pi/2,\pi/2)$ to $S^1$, e.g. $t\mapsto(\cos 2t,\sin 2t)$.

The resulting map $f:E\to S^1=:B$ is not a fiber bundle as a small neighbohood of a point on $\{p\}\times\mathbb R$ cannot have a connected inersection with a nearby leaf.

Replacing $\arctan$ by a function which converges to its asymptotic values sufficiently fast, one can make the map $f$ smooth (with zero derivatives at $\{p\}\times\mathbb R$).

Here is a counter-example to Q1. Consider a Reeb-like foliation on the annulus $E=S^1\times\mathbb R$. Namely fix a point $p\in S^1$, let $I=S^1\setminus \{p\}$ and fix a smooth function $h:I\to\mathbb R$ which tends to $+\infty$ at both ends of the interval $I$. One leaf of the foliation is $\{p\}\times\mathbb R$, and the remaining strip $I\times\mathbb R$ is foliated by graphs of the form $y=h(x)+const$. Note that all leaves are diffeomorphic to $\mathbb R$.

There is a map $f:E\to S^1$ whose fibers are leaves of this foliation. First define $g:I\times\mathbb R\to(-\pi/2,\pi/2)$ by $g(x,y)=\arctan(y-h(x))$. Extend $g$ to the entire annulus $S^1\times\mathbb R$ by setting $g(p,y)=-\pi/2$. Now we have a continuous map $g$ from the annulus onto $[-\pi/2,\pi/2)$ whose fibers are leaves of our foliation. It remains to compose it with a continuous bijection from $[-\pi/2,\pi/2)$ to $S^1$, e.g. $t\mapsto(\cos 2t,\sin 2t)$.

The resulting map $f:E \to S^1 =: B$ is not a fiber bundle as a small neighbohood of a point on $\{p\} \times \mathbb R$ cannot have a connected inersection with a nearby leaf.

Replacing $\arctan$ by a function which converges to its asymptotic values sufficiently fast, one can make the map $f$ smooth (with zero derivatives at $\{p\}\times\mathbb R$).