If one has a Belt of Physical Perfection; do your hit points go up?
If you have a Headband of Intellect; do you suddenly get more Skill points?
Both go up after 24 hours of wearing such items. Skill points granted by the Int increase should be noted separately from normal skill points to know which skills reduce after takig it/supression.
Do elemental damages stack? (as in could a person have a sword that did fire and cold damage all in the same round, or would it just do one or the other?)
Yes. I know that some people houserule flaming/icy properties to be incompatibile at the same time but there is no such limit in the rules itself. I prefer existence of "opposing" dualities and flaming icy swords.
If you had natural armour and an amulet of natural armour does it stack or do you just get the highest?
Yes. Amulet of natural armor grants enhancement bonus to natural armor improving it in a similar way as manufactured armors can have enhancement to armor bonus.