It's perhaps easier to point to instances of where air-barriers are absent. It's pretty common in some areas to discover that there is nothing but air between a tub-surround and naked batt insulation on exterior walls, leading to big air leaks/thermal bypass into floor or ceiling joists, etc. If you find something like that when you peel the exterior side, put SOMETHING in there to block the free passage of air.
Yes, you SHOULD be air-sealing all electrical & plumbing penetrations between stud-bays, as well as where they exit/enter the wall. This is more important when using fiber insulation, but even with open-cell foam the air seal isn't always perfect.
While there's a huge delta in environmental damage between EPS and XPS, it's not clear that iso is dramatically greener than EPS. Both EPS and iso are blown with pentane (~7x CO2 on greenhouse power) compared to XPS which is blown with HFC134a (at ~1400x CO2 greenhouse power.)
By
some estimates the lifecycle greenhouse potential per unit-R of EPS only 13-14% higher than that of polyiso, which is not a dramatically higher hit.
By comparison XPS has more than 500x that of iso (according to that source.)
Open cell spray foam delivers about ~44% more lifecycle greenhouse punch than iso, 26% more than EPS.
These are pretty squishy numbers, and it will vary by exact manufacturer & exact product, but in my mind open cell foam, EPS and iso are all pretty much in the same class, a class VERY DISTINCT from XPS or 2lb closed cell polyurethane spray, both of which use HFC blowing agents with high greenhouse gas potential. The chemical feedstocks for the different polymers vary somewhat, but there's nothing distinctly worse about styrene vs. polyurethane vs. polyisocynurate.
The biggest environmental hit with EPS is it's ubiquitous use in one-time-use "disposable" packaging or packing materials and it's low scrap value, resulting in a large amount of it in the biosphere (with a millennium long half-life in the environment). But that has little bearing on it's relative impact when used as a more permanent building insulation. (Iso is a more stable polymer, with a much longer half-life.)
Finding thin-stock standard densities of iso without foil facers is difficult (and you don't even want to know what unfaced or fiber faced 4lb density rigid-board iso costs!), but unfaced Type I & type II EPS is widely available in reasonable thicknesses.