Yes he did. Gollum told him so. That's how Sauron knew to look for the Ring in the Shire. After the Shire, almost all of Sauron's forces knew to look for Hobbits and to take them alive for questioning.

Yup. But Sauron did not expect a hobbit to be an adversary. That's a salient difference, in the context of Tolkien works.