To solve the strong CP-problem Peccei and Quinn suggested the use of a new $U(1)$-symmetry called the PQ-symmetry. For this symmetry they constructed an effective Lagrangian involving the Nambu-Goldstone-boson of the spontanoues broken PQ-symmetry:
$$\mathcal{L}_a := -\frac{1}{2}\partial_\mu a\partial^\mu a -\mathcal{L}_{int}(\partial_\mu a; \psi)+ (\frac{a}{F_a}\xi+\overline{\theta})\frac{g^2}{32\pi^2}F_{\mu\nu}^a\tilde{F}^{\mu\nu}_a $$
In his paper R. Peccei (https://arxiv.org/pdf/hep-ph/0607268.pdf) said the first and second term are needed to make the whole standard model Lagrangian invariant under $U(1)$ and the last term ensures that $U_{PQ}(1)$ has the right axial anomaly.
I have serveral questions to this arguments:
Why do we need the first and the second term to make the Lagrangian invariant under $U(1)$? Shouldn't it be invariant without that terms (beside from the anomaloues breaking)?
The standart model Lagrangian has already an anomaloues axial current from the QCD-sector. Why is it neccessary to implement the last term for this?
In which step of the solution to the CP-problem do we make use of the $U(1)$-symmetry? The mechanism is based on this effective Lagrangian couldn't it be constructed from another theory?