If you search for 'sha1 foo', that's being sent across the network to DDG's servers. And sure, if you're using SSL then it's not going across in plain text, but it's decrypted and handled on their servers in plain text; it'll probably even end up in logs and/or tracking databases somewhere. You're giving DDG your password.
At worst you're giving the attacker a hash target to try brunting. He still has to brute it, and that takes time. Select your plaintext from a large enough keyspace and it's astronomical time.
I'll need to review their policy more closely, but DDG claim fairly minimal tracking. At best someone might be able to correlate hash lookup with some IP space. That's a long way from handing over passwords. And as I already indicated, you could cradled the queries to make the search space much larger.
No, no, no. You're 100% completely misunderstanding this.
When you search for 'sha1 foo', that query ("sha1 foo") goes up to the server. They know your password is "foo" and that you're attempting to "sha1" it. They don't have a hash, they take that data and perform the hash, then send that down to you.