That I haven't been able to fix. The SDL library has a function that's supposed to set the taskbar icon, but nothing happened when I tested calling it on startup.
There is a workaround, though. First, to have the correct icon you must have Project Invincible installed.
Find the settings of your taskbar. (Where to find them depends on your desktop environment. Right-clicking the taskbar is usually a good start.)
Find a setting called Always use launcher icon or something like that and enable it. Save settings.
Actually resigning isn't even implemented. This is a bug in Arena 3.0. At least when having Project Invincible play on an ICS, when the engine announces a checkmate, Arena 3.0 interprets the announcement as resigning for the opponent. It is not allowed and causes Project Invincible to forfeit the game.
Unfortunately, the XBoard protocol specification requires that the chess engine announces checkmates. However, I have implemented a preference that allows you to disable these announcements and therefore work around the bug.
Start Arena and make sure that Project Invincible is loaded. Then go to Engines -> Engine 1 (or Engine 2) -> Configure.
A "Project Invincible Configuration" dialog will show up.
Uncheck Announce when the game ends.
Finally, click OK.