The stages are all easy and automatic except Advanced, which seems to hinge on your entire army taking a left turn on a 50/50 RNG pad...

The upgrade item for Mountain Lily is a character-specific skillbloom one that levels up her skill. The event page is correct it can be used on other characters but it seems to only give them exp, not skill levels. There's probably another item that unlocks her item slots, but I haven't seen one yet.