Question 1: Why would anyone who thought that they might not be paying all their US taxes use an exchange based in the USA? Is it something to do with needing to convert the Bitcoins to Dollars so that you can actually spend them?
Question 2: Given that one of the main selling points of Bitcoin is anonymity, why would someone operating an exchange keep any but the barest records? I appreciate that they can't destroy the information now they have been asked for it, but I am trying to grasp why they would put themselves at risk of being in that position by retaining it in the first place?
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Easy, Coinbase is one of the least sketchiest exchanges around. Given the amount of personal information required (see below) you generally want one that won't go bankrupt overnight (like a certain Magic the Gathering Online Exchange).
Also, they are one of the easier ones to use.
As for your second question - well, you need quite a bit of personal information - at least a name and address if you want to be able to pay someone, banking information if you want to electronically transfer funds. And I believe you can even use debit to buy bitcoins, which requires a bit more information to ensure it isn't fraudulent.