Running the game in windowed mode is the best way to run the game AND check the net for stuff, view image charts, etc.
Otherwise the game just closes/crashes. (This is true on the 4 computers with various operating systems I've tried it on)
You always get booted back to lobby after finishing a quest, so you'll have to get used to remaking groups. (sorry! xD)
And Cyan's link will take you to http://www.phantasystaronline.net(or ULTIMA's main page, not forum) where you can view server status on the Right-hand side of the page.
Also, if you run the game with the Launcher that comes with the game, it will tell you before you boot the game up if the Server is up or not and how many players are on!
=)