Add documentation for cratchit

We should add code-level documentation to cratchit so when we publish to crates.io, the documentation gets published as well.

Additionally, we should add a badge to indicate the state of the docs.

Start by reviewing the Rust crate's public code and its crates.io publication setup. Add the requested code-level documentation and a badge showing the documentation state; done means the documentation is published with the crate and the badge is present.

Written by the indexing model from the issue text.