This is explained pretty clearly in Cisinski and Deglise’s book. The input from Ayoub’s thesis is the purity property for the projections $\mathbb{P}^n_S \to S$, see Theorem 2.4.28.

The way they achieve this is nothing too surprising; they simply follow Deligne’s original strategy in SGA4 using compactifications to glue along the case of open immersions and proper morphisms, instead of Ayoub’s approach using closed embeddings and smooth morphisms (which only exist under some hypothesis like quasi-projectivity).

Regarding the second question, see Proposition 2.3.11 in their book.

This is explained pretty clearly in Cisinski and Deglise’s book. The input from Ayoub’s thesis is the purity property for the projections $\mathbb{P}^n_S \to S$, see Theorem 2.4.28.

The way they achieve the generalization to non-quasi-projective schemes is nothing too surprising; they simply follow Deligne’s original strategy in SGA4 using compactifications to glue along the case of open immersions and proper morphisms, instead of Ayoub’s approach using closed embeddings and smooth morphisms (which only exist under some hypothesis like quasi-projectivity).

Regarding the second question, see Proposition 2.3.11 in their book.