No.

Not unless the description of the magic standard in question states that the weapons of a unit carrying it gain the Magical Attacks special rule. Generally speaking, purchasing a magic item doesn't conjure up additional bonuses from the æther.

Yes. The wearer of the Wizarding Hat is not actually a Wizard – their magical powers are granted by a haunted hat which is not affected by any armour the model may wear.

Yes, because one is an enchanted item, the other is a type of magic armour, but we hope anyone equipping a model in such a way would make the effort to depict it wearing two hats!

No. As stated in the item's description, the wearer of the Wizarding Hat is a Level 1 Wizard and knows one randomly generated spell. This is clearly a huge detriment to a powerful Wizard, but might be a boon to any character that is not a Wizard.

No.

The Wizard is transmogrified, but not their mount. In the example given, you would be left with a Frog riding a Dragon.

No.

An Arcane Familiar does not increase the number of spells a Wizard knows, it simply allows them to know spells from two Lores. Thus a Level 3 Wizard would know three spells from two Lores.

Models can purchase magic items from the list of common magic items in the Warhammer: the Old World rulebook, or from their own faction's list of magic items. A Dragon Ogre Shaggoth, for example, can purchase items from the common or Beastmen Brayherds magic item lists.