Question:  If you have a magical weapon like a Sword of Certain Death, that says an enemy hit with this weapon can't regain life until your next turn, but have made the sword your arcane focus with the Pact of the Blade invocation Improved Pact Weapon, does the effect get transferred to your spell attacks made through the weapon as well, because those are technically "attacks" made with the weapon?

Nope. Just as the sword’s bonus to attack doesn’t apply to spells you cast with it as your focus, any other special functions don’t apply.