Lorentz invariance, locality and unitarity of the S-matrix

Question

I am reading these notes on cosmological bootstrap by Baumann and found the following statement, which I am trying to understand / see where it comes from [page 2 of the linked notes]:

"[...] S-matrix bootstrap, where the structure of scattering amplitudes is fixed by Lorentz invariance, locality and unitarity:

$$A(s,t) = \sum a_{mn}s^n t^m + \frac{g^2}{s-M^2} P_S \left( 1+ \frac{2t}{M^2}\right)$$ "

Essentially he says that Lorentz invariance, locality and unitarity fix this form of a scattering amplitude. I am asking myself three things regarding these three aspects:

1. Lorentz invariance: Shouldn't this imply that $A$ depends on lorentz-invariant quantities only, i.e. $s,t,u$ and not only $s,t$?
2. Locality: Why does locality imply that the only poles appear in internal propagators? (similar, currently unanswered question here)
3. Unitarity: In the notes he connects the $g^2$ with unitarity, but I cannot see the connection here. What does unitarity exactly imply in this context?

Thank you in advance!