I'm also stuck connecting to Trinity Core, with the Connected loop issue. I can authenticate just fine (and it fails if I put in bad credentials), so I'm definitely connecting to the Auth service from the client.
The issue seems to be connecting to the worldserver, most likely from some sort of misconfiguration through the networking side of things. It will show the Trinity server as an available realm even if I don't start the worldserver service, so there is definitely some sort of disconnect there. I've also tried setting the address to 127.0.0.1 instead of the LAN IP to the same effect. The worldserver is connecting to the database just fine, apparently. If I try to change any of the db connection strings it crashes immediately. If I try to change its port it crashes when trying to initialize the network. I'm not much of a networking person so I imagine there is something there that I missed. I currently have the ports forwarded and have the firewall turned off.
As a side note, I installed and built this from your instructions using the latest files on Windows 10 with a 64 bit installation. I thought the instructions were really good up until the middle of "Server Setup" and then they got a bit hairy. You have to jump around a bit between Server Setup, Database Installation, Client Setup, and Networking, whereas before it was very linear. For example, Server Setup has a step that says "Log in using the user/pass you just created." but you don't create a user/pass until the Networking step, two sections later (and those instructions are a bit vague as well, not telling you *where* to enter the commands).