Homomorphic Encryption - Smart Vercauteren Batching
Yes, technically this is an isomorphism, so $\cong$ should be used. The isomorphism is question identifies $\theta$ with $X$.
Not quite (for the original question or your restatement).  The notation $\mathbb{Q}(\theta)$ means adjoining $\theta$, a root of $F(X)$ in the algebraic closure of $\mathbb{Q}$ (namely, $\mathbb{C}$), to $\mathbb{Q}$.  The finite field $\mathbb{F}_2$ has nothing to do with it.  (And I think $\mathbb{Q}/\mathbb{F}_2$ is meaningless, since there is no field embedding of $\mathbb{F}_2$ into $\mathbb{Q}$.)
Yes.  Such a structure is more commonly known as a number field (i.e., a finite dimensional field extension of the rationals).