Yes, Stonehearth must be closed before trying to join a multiplayer game. When you click Join Game on the Steam Friends list, it will automatically open and show the connection screen, which lists any mods from the host, and allows you to continue if both players have the same version and mods.
What do you mean your Steam name was missing and there was no Invite button? We had some issues about this in the past, but it was a bug with private profiles in Steam, and they fixed it eventually.
If you had Steam running, and already started a multiplayer game and placed your banner, then you can go to your Steam Friends list and right click on your friend’s name and click on Invite to Game (or your friend can right click your name in his friends list and click on Join Game). Always after the host has gone through embarkation and placed the banner.