The guide is now complete! Thanks everyone for your help!
Pink for the Weekend - Log in on Friday, Saturday, and Sunday (CONFIRMED)
Out of Space - Run out of space in your etc. inventory (CONFIRMED)
World Famous - Get 101 fame (CONFIRMED thanks to AlexF)
The Pink Party - Be in a party of 6 Pink Beans (CONFIRMED thanks to BaoDovvn)
Inhuman Lie Detector - Use 30 runes. Quest name is a mistake. (CONFIRMED thanks to ZeroByDivide on reddit)
Narcissist's Cosplay - Use a Pink Bean transformation potion from a 12th anniversary box (CONFIRMED)