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Coal mines and cloth location:
The villagers are in the upper northeast of the mines behind crackable walls. The walls are a different brick than the surrounding walls. You need a Miner's Axe to break them. You can buy that from a blacksmith that sells normal weapons like Hodge the Smith.
Dungeon Diver sells the Sword of Death that you need to kill the creature but the chances of it's appearing in his inventory is 1 which means you may have to sleep in an inn or camp out outside his store until it shows up but eventually it will. Orc the Armorer also sells it.
Liz sells the Cloth you need. She has a bookstore in the Astrologer's Guild and sells soft cloths, the kind to dust off books. Same cloth will due for Madame Repulsa to give you a better sword.
To kill Otho and his hound you need to smash the spheres. To do that you need Mage Otho's Rod.
Fire Mushrooms and Coat of Arms:
Joint Eater is a very tough monster. Coat of Arms- red bush in the south need Dexterity over 20. From statue you need Personality greater than 23. My guess is those are your missing pieces. Fire mushrooms you can buy in an alchemy shop that sells normal ingredients.
Gavina should sell your Fire Mushrooms. You might have to camp out.
If your Endurance is greater than 25, you get a Shield Piece in the north on a fern.
If your Strength is greater than 30 you get one from a haystack.
If your Intelligence is greater than 20, you get one from a red bush in the mid north.
You get one from a rock in the mid west if your Strength is greater than 31.
You get one from a swan if your Personality is greater than 20.
Use the spell Abyssal Rage on Largra.
Look for the harpist's voice in the Boiling Loch.
Finding the Moon Goddess: You need some water. Look near the pillars in Three Sisters in the northwest at night. You have to find the right tree.