I have been looking into this issue. I have all the same issues trying to log in. I just came across a blizzard response regarding the server issues. In the article I came across this: : " To diagnose and address them swiftly, we need to make sure the “herding”–large numbers of players logging in simultaneously–stops. To address this, we have people working on a login queue, much like you may have experienced in World of Warcraft. This will keep the population at the safe level we have at the time, so we can monitor where the system is straining and address it before it brings the game down completely. Each time we fix a strain, we’ll be able to increase the population caps."
So in summation it seems that Blizzard is capping how many players can be online, queueing everyone else, then dropping you and sending you to the back of the line if your turn comes to log in and there is already the capped number of players online. Such is my theory anyways. Link to the Blizzard response here. Diablo II: Resurrected Outages: An explanation, how we’ve been working on it, and how we’re moving forward