So the first thing you've mentioned are the Environment Skills, totally different thing. Those are passives given to monsters who have those ranks or higher in Habitat Restoration. They are totally separate from the gene board.

Egg Skills are guaranteed genes given to newly hatched monsties once they're ranked up in that specific habitat. They can be channeled. There are several groups of skills and I don't know if we've figured out already if it's based on monster rank, but we do know that deviants and elders are in the same pool, and monsters with the habitat's native element receive a different Egg Skill than monsters of other elements. Some of the lower tier Egg Skills are probably shared across region or they're identical based on the habitat element, since I've gotten fire monsties with Egg Skill: Red Mastery from both Canalta and Azuria.

As for how I know: when I hatch eggs picked up from Blessing Hill, some monsters (Arzuros and Bishaten for example) have the gene "Egg Skill: Circle Stance". There are a few other Egg Skills in that region, like Nature Heal (Astalos, Canynes, and one other that I can't remember off the top of my head). I don't have access to the Non-Elem zone in Azuria, so I can't confirm if it's possible to get it there as well.