You do not need to run the Host with a shortcut, it isn't supposed to work that way. When you install the Host with default settings its service should automatically with Windows. If it doesn't, you should figure out why , this knowledge base article can help. You can also manually start and stop the Host service by using shortcuts "Start Host" and "Stop Host" found in the Windows Start menu.
However, when you directly click on rutserv.exe file in the Program Files folder you run the Host in the Agent mode. This behavior is a side effect of the current implementation and we will fix that soon.
This is not only an application password, perhaps the wording in the interface is misleading a bit. This is also address book encryption passphrase. If you forgot the passphrase, you cannot unlock or otherwise restore the address books because they remain encrypted.
You can, however, unlock the interface if you delete config_4.xml file located in the Viewer application folder (%APPDATA%\Remote Utilities Files\ ) . Note, that the Viewer won't be able to load your encrypted books (because they are encrypted on disc) and will create a new, empty default address book for you.
Hope that helps. You can learn more about how address book encryption works by following the link above.
I think this is not about the end points, it's about the router. You might want to check with your ISP if your new router supports NAT loopback, or just generally describe the problem to them. They should be able to help because they know their known infrastructure better.
If you want complete control over the remote computer including the ability to control UAC prompts and install applications, you should use Host instead of Agent. Unlike Agent, the Host works as a system service with system-wide privileges.