1. For which Hand was dug the passage to Chataya's?
No comment. Feel free to theorize...
2. How could Lady Joanna Lannister have been a childhood friend of Prince Doran's mother? Doran is 50 in ASOS and Tywin, what, 57-58? Was he significantly younger than his wife?
The queen's companions weren't necessarily all of an age. Take a look at the group of ladies who attend Margaery Tyrell, for instance -- a mixture of girls her own age and some who are notably older and younger.
No, Lord Tywin was not significantly younger than his wife.