Most character greets tend to occur as listed in MDE. I think they open at official park opening. I don't tend to focus on character greets during early entry or rope drop.
In the past, WDW periodically has offered surprise characters. They tended to happen on very busy weeks, like Easter and Christmas. When they happen, one location has been Main Street. I have sometimes seen characters out early in the morning, but I can't say 100% if it was EE or at park opening.
Another time surprise characters have happened in the past- at least from what I have seen/read was that characters sometimes made surprise appearances for training purposes. I don't think they tend to be at rope drop, but more middle in the day. One location has been the Italy Pavillion, featuring rare/unusual/lesser-known characters. I've seen them out a few times. I've also sometimes seen characters show up mid-day near park exits (AK,MK) that may also be surprise appearances, but I didn't check MDE to verify.
Earlier this year, Mandolorian was making surprise greets in GE. We happened to catch him, and it was neat. It was mid-day, in an out of the way nook in GE. He appeared about the same time Storm Troopers were walking through the main area of GE. I have not paid attention to know if they are on-going or if they are now announced in MDE.