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In my opinion, the proposed hero mage build has too few passive skills. Why so many spells? I absolutely agree that summoning dark and earth elementals should be added. However, "Flaming Resolve" is questionable – increasing morale isn't the most important thing. If the opponent is a mage, they will counter with something more sinister, such as petrification on champions. And if the opponent is a warrior, the focus should be on disabling enemy units rather than speeding up your own. You can also do perfectly well against neutrals without this skill.
I would add a passive skill to increase defense. Otherwise, the units will be too fragile. I would remove "Stone Skin" altogether. It's not needed against mage heroes, and it won't help much against warrior heroes – it's more important to eliminate individual, highly effective stacks from the game. You can also clear neutrals perfectly well without this skill. Petrification and Fear are what you need to use in battle.
"Drain Life" and "Regeneration" are both general healing skills. It's tempting to have both, but I would choose one, specifically "Regeneration," as "Drain Life" is useless against undead. Casting both takes too long. In that time, the opponent will definitely do something more unpleasant. It's better to put these skill points into a passive skill or elementals.