Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the form $Q[\theta]$).
Can someone give a confirmation, reference, proof-sketch, hint?
And is there a similar result for $Z$ ?
You are looking for the primitive element theorem. A quick Google search will give you several different sets of self-contained notes that include the proof.
If your second question is asking whether given a collection of algebraic numbers $\alpha_1, \dots, \alpha_n$, there exists a $\theta$ such that $\mathbf{Z}[\theta] = \mathbf{Z}[\alpha_1, \dots, \alpha_n]$, then the answer is no.
For a concrete example, let $\alpha$ be a root of $x^3 - x^2 - 2x - 8$. Then for any $\theta$ (necessarily in $\mathbf{Q}(\alpha)$), we have $\mathbf{Z}[\theta] \neq \mathbf{Z} \left[\alpha, \frac{\alpha^2 + \alpha}{2} \right]$, the latter of which is the ring of integers in $\mathbf{Q}(\alpha)$. All of this can be proved via methods from basic algebraic number theory.