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The ghost appears after midnight. To talk to her, you need to equip the spirit stone.

But you also need to stay awake after midnight, so you need to make and use an energy potion to stay awake.

Energy potions are made after you have crafted a potion or two. You'll need an energy mushroom found in the forest for that.

When you fight the ghost, let her win again. The story should progress after that. 

It sounds like you're still pretty early in the game. I'll try to help you as bests as I can remember since it's been a while. 

For the cat:  At some point you open up the jungle-like area south of the forest, called Palmtree Spot or something. It's the zone with the jellyfish that can paralyze. Explore that area until you find the Fisherman's dock. Help the fisherman by giving him one of your items (hint: you mentioned the item that you need in your post). He will give you something to feed the cat. 

For the ghost: This one I don't remember exactly. I know that early on the ghost only appears at night (try after midnight. You might need to use an energy potion to stay up long enough). Early in the game you also need to have the spirit stone equipped to see the ghost. If you don't have the spirit stone yet, talk to Evie (I'm pretty sure she's the one who gives it to you).