Where to find a kerosene lamp, potatoes, and a couple of disks - visit Lina in the morning in her room and ask her to go with you to the warehouse, then talk to her again 

Gatekeeper answers - shadow, tree, 5

Liibrarrian answers - seth, iris, min

Warrior skills guide - complete Lina and Aeria arc in the anime world:

Find the other way to beat them - learn 'fire acceleration' from Imira(it requires 60+ fire resistance), and  'shadow step' from Klio

How to help Fiona decipher the scroll - wait until Cyberworld is unlocked(day 20+) and talk to Iona-3(hologram) there

Where to find a screwdriver - check boxes in the basement a few times

Where to find the door for the key you found on the new island - click on the hatch in the voyeurist's tower