Yes, I ran them early in the morning because we were going to parks, but you could really run these routes at anytime.
There is a sidewalk all along Epcot Resorts Blvd except for a small portion just before and after the viaduct (between Beach Club Villas and Boardwalk). It's only about 50 feet or so between the back side of Beach Club Villas and the start of the viaduct sidewalk, but it's a little longer, maybe 150 feet, on the other side of the viaduct before getting to a sidewalk by the Boardwalk (although there is some space in the grass on this stretch if needed).
For the longer routes with the loop around DHS parking lot, I stayed just inside the parking lot barriers/poles, so that I was separated from the cars entering DHS. It wasn't bad at all.
ETA: when I say viaduct, the road becomes a tunnel that goes under the boat viaduct. In any case, there is a sidewalk when you go under the viaduct, just not right before and after.