Do due dates and completed tasks carry over to Asana?

Yes to both. A task's date arrives as the Asana due date (Asana dates are day-based, which matches how dates work here — no times to lose). And a push includes the task's current state, so a to-do you'd already ticked off lands in Asana already marked complete. As with every app, the hand-off is one-time: after the push, completing or re-dating the task on either side doesn't change the other.