Edgar must be under your control in order to start casting an artifact spell using its first ability, but it doesn't need to stay under your control throughout that process or be under your control as the spell resolves. If, for example, you sacrifice Edgar to pay an additional cost of the spell or Edgar is destroyed in response, the spell will still resolve normally.

If an artifact you control has {X} in its mana cost, X is 0 for the purpose of calculating its mana value.

You must still pay all costs and follow any timing restrictions and permissions for artifact spells you cast from your graveyard this way. Normally, you'll be able to cast them only during your main phase while the stack is empty.

The value of X is calculated only once, as Edgar's last ability resolves.