If I play a dwarf wizard, does he have both Wizard weapon proficiencies (e.g. daggers) as well as Dwarf weapon proficiencies (e.g. battleaxes)?
Yes, the Dwarf Gains Proficiency with Dwarf Weapons and Wizard Weapons.
Unless traded for an alternate racial trait, the dwarf gains the Weapon Familiarity offense racial trait, which reads
This doesn't alter or replace the weapon proficiencies granted from taking a level in the wizard class. As such the typical dwarf wizard is proficient with the battleaxe, heavy pick, and warhammer because the character's a dwarf and the club, dagger, heavy crossbow, light crossbow, and quarterstaff if he's a typical wizard. The typical wizard doesn't gain proficiency with any martial weapons, so the latter part of the Weapon Familiarity racial trait is unaffected.