An item is allowed to have additional magics added to it, where the cost of the addition is the cost of the final product minus its current value. That is, take a +1 quarterstaff (2300 gp) and make it a +2 quarterstaff (8300 gp) for 6000 gp. See Creating Magic Items – Adding New Abilities:
Adding New Abilities
A creator can add new magical abilities to a magic item with no restrictions. The cost to do this is the same as if the item was not magical. Thus, a +1 longsword can be made into a +2 vorpal longsword, with the cost to create it being equal to that of a +2 vorpal sword minus the cost of a +1 sword.
If the item is one that occupies a specific place on a character’s body the cost of adding any additional ability to that item increases by 50%. For example, if a character adds the power to confer invisibility to her ring of protection +2, the cost of adding this ability is the same as for creating a ring of invisibility multiplied by 1.5.
For many items, combining functions (rather than upgrading them as with weapons) in one item carries a price premium equal in value to half the value of any functions after the first. This would also be part of the cost of adding magic to an already-magic item.
Stuff like directly using hides or medusa heads to make items, kind of. By default, yes, your character needs to have the appropriate materials (whose worth equals the crafting cost of the item), but what the “appropriate materials” are is left undefined and up to the DM.
Thus a DM may decide that a spike, gem, etc. is worth a certain amount towards the crafting cost of the item (e.g. that sword costs 2000 gp to make, but the gem you just found can be used for a part of it: you only need to provide materials [steel, leather, whatever] worth 1000 for the remainder of the sword). Most of the time, though, it’s more like “I go to the blacksmith and buy whatever I need for the sword from him.” Requiring specific, especially rare, materials doesn’t really occur in the rules except sometimes for certain spells (e.g. raise dead requires diamond dust).
I haven’t personally seen one, but I’d be surprised if no one’s drawn up lists of explicit materials for particular items. I doubt it’s official, though.