Tha answer to question 1 is negative. There are Frechet spaces $X$ whose strong duals $(X',\beta(X',X))$ are not barrelled (equivalently, not bornological by a theorem of Grothendieck). The first examples of such non-distinguished Frechet spaces were constructed by Köthe and Grothendieck but there are also examples which are very easy to describe: According to Taskinen the Frechet space $C(\mathbb R) \cap L^1(\mathbb R)$ of continuous Lebesgue-intergrable functions (endowed with the seminorms $\int|f(x)|dx + \sup\{|f(x)|: x\in [-n,n]\}$) is not distinguished. This answers question 1 with $E=(X',\sigma(X',X))$.

The answer to question 1 is negative. There are Frechet spaces $X$ whose strong duals $(X',\beta(X',X))$ are not barrelled (equivalently, not bornological by a theorem of Grothendieck). The first examples of such non-distinguished Frechet spaces were constructed by Köthe and Grothendieck but there are also examples which are very easy to describe: According to Taskinen the Frechet space $C(\mathbb R) \cap L^1(\mathbb R)$ of continuous Lebesgue-intergrable functions (endowed with the seminorms $\int|f(x)|dx + \sup\{|f(x)|: x\in [-n,n]\}$) is not distinguished. This answers question 1 with $E=(X',\sigma(X',X))$.

EDIT: If $M$ is a barrel in $E$ then its polar $M^\circ$ is a $\sigma(X,X')$-bounded subset of $X$ and hence it is bounded in the Frechet topology so that $M=M^{\circ\circ}$ is a $\beta(X',X)$-neighbourhood of $0$. Conversely, the typical $0$-neighbourhood $B^\circ$ in $\beta(X',X)$ with a bounded subset $B$ of $X$ is a barrel in $(X',\sigma(X',X))$.