Check out how to use millis()
That will give you a time reference in milliseconds. From there you can detect your events (day, 10s forward, 5s pause, 10s reverse).
Being a first project, I would set it up so that when you turn it on the door sequence runs. Call that 0 hour at 24 hours later it will do it again. You can get more complicated using a real-time clock, but that is way more complicated.
Program it up using millis() and show us what you have or where you may be having an issue.
P.S. An Arduino Mega ADK is way more power than this needs. An Uno or Nano can also handle this at much less the cost.