If another player takes control of the Factory while it is animated, that player keeps control until the control effect ends. They do not lose control just because the card stops being an Assembly-Worker. Note that Auras which steal cards will go to the graveyard if the permanent they enchant is no longer valid.

A noncreature permanent that turns into a creature can attack, and its abilities can be activated, only if its controller has continuously controlled that permanent since the beginning of his or her most recent turn. It doesn’t matter how long the permanent has been a creature.

Activating the ability that turns it into a creature while it’s already a creature will override any effects that set its power and/or toughness to a specific number. However, any effect that raises or lowers power and/or toughness (such as the effect created by Giant Growth, Glorious Anthem, or a +1/+1 counter) will continue to apply.

Once turned into a creature, Mishra’s Factory’s last ability can target itself. If it’s already blocking, this won’t remove it from combat.

If you turn Mishra’s Factory into a creature but haven’t controlled it continuously since your most recent turn began, you won’t be able to activate its first or last ability.