3. It would be almost trivial for the U.S. government to snoop on DuckDuckGo's traffic at the switch or cloud provider level even without informing DuckDuckGo of this. I believe they host at EC2 East, which is in DC and a major U.S. government linked hosting provider (see Amazon FedRamp).
Oh, I'm sure they can easily collect the traffic. I think any secure service needs to be made with that in mind now. But the question is if they can get the data unencrypted.
The cloud is not private, period.