Spears cause Piercing damage, so your damage when using this will be piercing.
And I assume, that if your a warrior secondary, Strength will still increase your damage output even if using a spear. It's highly flawed if it does not.
The AP from strength works on attack skills only, and yes, it does work for spears.
Strength is the primary attribute, so you can't use strength as a secondary warrior.
I don't think there is any attack skills that can be used with a wand/staff, so unless I'm wrong about that, strength can't effect wands/staves.