1) If $E/F$ is a Galois extension and $B$ is an intermediate field, then $E/B$ is a Galois extension.
If $E/F$ is a Galois extension, then $E$ is a splitting field of some $f(x)\in F[X]$. Can I just say that $F\subset B$ and so $f(x)\in B[x]$ and thus $E$ is a splitting field of some $f(x)\in B[x]$, therefore $E/B$ is a Galois extension ?
2)Show that being Galois need not to be transitive; that is, if $F\subset B\subset E$ and $E/B$ and $B/F$ are Galois, then $E/F$ need not to be Galois.
Hint: Consider $\mathbb Q\subset \mathbb Q(\alpha)\subset \mathbb Q(\beta)$, where $\alpha$ is a square root of $2$ and $\beta$ a fourth root of $2$.
By the Hint I take $F=\mathbb Q$, $B=\mathbb Q(\alpha)$ and $E=\mathbb Q(\beta)$ whith $\alpha$ a root of $X^2-2$ and $\beta$ a root of $X^4-2$, but How can I conclude ?
1) Your argument is correct, but you should say something about separability as well.
2) Any extension of degree $2$ with characteristic $0$ is Galois. On the other hand, $\mathbb{Q}(\beta)/\mathbb{Q}$ is not normal, since $x^4-2$ does not split over $\mathbb{Q}(\beta)$.