To unlock Champion, you need to clear the 3* version of the dungeon you're playing. So to get Champion Whisper Woods, you need to clear 3* first.
As for the mission, it is not solo only so you can party with others to get the drops. The drops are independent for each player, meaning that everyone has a chance of getting one if they have registered the mission.
For the dungeon itself, it depends on how high of a level you have gotten and/or on how adapt you are at using your best character in dungeons. Nothing is terribly hard to defeat. The Ruins of the Silver Knights boss may be somewhat irritating since they teleport around but that's all I can think of.