Q:

Do Zombies with skulls drop the skulls? 

I have been playing Terraria for a while and I have always seen zombies with skulls on their heads.  I think the skulls are cool so I have always wanted to wear one of those.  So I need to know, can those Zombies drop their skulls? 

A:

There are no zombies with skulls on their head.  What you are probably referring to is a swamp zombie.  And although it looks like a skull its not, all zombies have a chance to drop a shackle, a zombie arm, and a zombie banner. 
Some zombies also have their specific drops :
Those you can encounter in a regular night :

Slimed zombie : drops gel
Pincushion zombie : drops arrows

Those that appear commonly inside biomes, or events. 

Rain zombie (during rain) : drops Rain coat, Rain hat. 
Zombie eskimo (surface snow biome) : drops eskimo armor
Spore/Mushroom zombie (hardmode mushroom biome) : drops significantly more coins
exception : Disguised zombie (halloween) : doesn't drop anything but little money. 

Those that appear rarely

Doctor bones (jungle biome) : drops his Archaeologist's hat
The groom (blood moon) : drops his top hat

