eh still no answer.. ok
1)Earth Enchantment is %fire.. brimstone is %physical.. %fire boosts any flat fire damage you have both from spells and weapon attacks.. yes it will buff the core dweller as well
2) what do you mean by active skills? like VO, eruption, flame surge? if so, no
3) since AE, %ele damage now boosts ele DoT
advanced tips: it bothers me that it seems you are planning your char to be spells + staff attacks.. it only works very early game but later on if you push with that strategy, it's a recipe for disaster.. staff attacks become laughable later on and since you also need life steal and high attack speed for that.. uhhh just no.. just go for CDR and cast speed.. all this changes if you pick spirit as 2nd mastery (ternion)
were you perhaps planning on buffing your Core Dweller as well as your spells? ughh no.. one has to go.. I'm not saying you shouldn't put any points on Core Dweller, i just mean put a few points on it but don't buff it with pet items.. except if you want a dedicated petmaster.. for that you equip pet jewelries (preferably 3) and sacrifice mage items