What do the icons next to defeated opponents mean

I beat this using Chandra's Unquenchable Fire for all three rounds. Cast your damage spells directly on the opponent when possible. Also, your creatures should act as shields, so try not to use them too offensively.

Just keep using instants and sorceries to win each round. Throw in whatever cards you can to make this happen.

The trick is just having enough creatures between you and your pal to survive. Don't try to overpower Garruk's creatures, he can pop out way too many--that is what green is for, after all. Just go straight for him.

Also, if you bump the difficulty up, it might make the enemies a bit harder, but your ally will play better as well.

The invites in MtG:DotP don't always pop up for your friends like they're supposed to. However, your friends can still join your game: In the steam friend's list, they need to click the arrow next to your name, then click "join game." They can bring up the friend's list ingame by hitting shift + tab, or simply by minimizing the game (alt + tab) and opening it from Steam.

You can do the same thing to manually invite friends using your friend's list.

Note that joining and inviting to games both work for most multiplayer Steam games, not just Magic the Gathering.