Rope drop is just the name given to the time when they start letting people into the parks, which tends to be about 10-20 minutes before the official posted opening times. When someone says for you to be at the parks at rope drop, they are really saying "be there early enough that when they start letting people in before official opening you are there". Nothing special, just getting there early enough to be in the first wave. Also, being at MK early means you get to see the opening show, which as I understand it happens every morning. I don't think the other parks have opening shows, though they might have some characters out to welcome guests.
What makes this tricky is that some days one of the parks will have Extra Magic Hours(EMH), which is where people staying at Disney resorts get in an hour before everyone else. So if you ARE staying at a Disney resort and you get to the EMH at their rope drop, then you will have a very easy time seeing lots of stuff before offsite guests can enter. If you are an offsite guest, unable to attend EMH, then you are at a disadvantage those days because you cannot get in with the first rush of people. EMH do move from park to park as you mentioned in your first post. So Monday could be MK, Wednesday could have EMH at HS, and then Thursday they could be at EP.