The effect making the creature a Phyrexian doesn’t have a duration. It lasts until the creature leaves the battlefield.

If the creature stops being a creature after the ability has resolved, it will also stop being a Phyrexian until it becomes a creature again, if applicable.

Terror of Towashi’s triggered ability triggers and goes on the stack without a target. If you pay , the second “reflexive” triggered ability will trigger. You’ll choose the target creature card for that second ability at that time.