No, it does not hold in general, for example take $P=x^{3}-2\in\mathbb{Q}[x]$
: $[L_{P}:\mathbb{Q}(\sqrt[3]{2})]>1$ since $L_{P}$ have elements
that are not in $\mathbb{R}$, but $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$
hence $[L_{P}:\mathbb{Q}]>\operatorname{deg}(p)=3$ .
For your second question the answer is almost: If $g(\alpha)=0$ then
$m_{\alpha,F}(x)\mid g(x)$ but since $g$ is irreducible then you
have that $m_{\alpha,F}(x)=cg(x)$ for some constant $c\in F$ (otherwise
you would've had a decomposition of $g$ ).
The minimal polynomial is defined so it is monic i.e s.t that the
leading coefficient is $1$ so $g$ divided by its leading coefficient
is monic and equals to $m_{\alpha,F}(x)$.