We attempt Rope Drop nearly every morning of our trip.
We try to get to the MK early enough for the opening show & try to get to DHS at least 15-30 minutes before the park opens. At EPCOT & AK, we normally arrive around 9:00 am & are going through the turnstiles usually at 9:00ish. (At AK, it's sometimes later.)
The day we are going to the World Showcase is our sleep-in day, so we don't arrive to EPCOT on that day until sometime around 11:00 am.
Last year, on our first DHS day, we just couldn't get moving & didn't get inside the park until around 9:30 am - OMG!! We were lucky to get fastpasses for Toy Story Mania! On our 2nd DHS day, we arrived around 8:30 am. What a difference! We were able to ride Toy Story once, stand-by, & then also get fastpasses for later.