Almost all animals on materials components only say the animal's name, without stating if it must be alive or dead... with one exception. The Infestation spell requires a living flea"living flea".
As per rules on Material Components:
Casting some Spells requires particular Objects, specified in parentheses in the component entry.
You could interpret this in one of two ways:
a) Having (living) in one animal means this is the only exception. The others are dead and treated as objects for this purpose.
b) Having (living) in one animal means the others can be either alive or dead. This would mean the description of material components being objects is a just a overlook on the writer's part.
Personally I believe a is the right one.
#If the monster could be used as component
Having spider as a component doesn't mean anything that resembles a spider can be used. A _Giant Spider Wizard_ is not a Spider.
It's also important to remember that materials without a specified cost are not consumed unless the description says otherwise. So the spider wizard would not be consumed by the spell in the case of spider climb.
