2. Even if you register your real name, address, phone number, etc. with an exchange, why can't you send them overseas to a foreign exchange, where your local "authorities" have no jurisdiction?
If you're trying to evade the US government, and they care enough, it will be hard. They have a lot of influence. They can pressure other governments to do things. If Ross Ulbricht had mixed his coins using Bitstamp, do you think the US government would have given up?
If you are trying to evade a government without a lot of influence, or are only mixing small amounts of coins, then maybe using an exchange that requires ID is good enough.
Exchanges that don't require any ID should be safer, but if they keep logs of their incoming and outgoing transactions and associate them with an account, then you could still be traced if your government pressured them into revealing the logs.