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 $m_{\alpha,F}(x)$.