Do Bracers of Armor Stack with Armor?

Bracers of armor and ordinary armor do not stack. If a creature receives a larger armor bonus from another source, the bracers of armor cease functioning and do not grant their armor bonus or their armor special abilities.

Does Mage armor stack with bracers of armor?

Yes, they do stack

As the replies to his tweet point out, the description of the Bracers of Defense says "you gain a +2 bonus to AC"; they don't provide a different AC calculation, so the two do stack.

Does ring of protection stack with bracers of armor?

Yes. Although each provides a bonus to AC, each item's type of bonus is different. Bracers of armor provide an armor bonus, amulets of natural armor provide a natural armor bonus, while rings of protection provide a deflection bonus.