Is there a degree-2 SOS rewriting of the Goemans-Williamson rounding?
The quadratic sum-of-squares relaxation of Max-Cut should indeed work out to be equivalent to the SDP that Goemans and Williamson solve. I encourage you to work this out yourself. This means that the Goemans-Williamson analysis indeed shows the inequality you mention.

You ask whether degree-2 SoS is a rewriting of the Goemans-Williamson rounding. It isn't. Rounding is a completely separate issue. Degree-2 SoS is a rewriting of the SDP that Goemans and Williamson solve. Rounding is how we bound the integrality gap.

Assuming the Unique Games Conjecture, $\alpha_{GW}$ is the optimal approximation ratio achievable in polynomial time. Since optimizing degree $d$ sum-of-squares can be done in time $O(n^{O(d)})$, this implies (assuming UGC) that every constant degree $d$ has the same integrality gap $\alpha_{GW}$ in the worst case.