Refactor: Use doctate-common for shared logic

Moves shared types and utility functions to a new `doctate-common`
crate.
This includes:
- API key header constant
- AckResponse and AckStatus types
- Timestamp formatting and parsing utilities

This change centralizes common functionality, reducing duplication and
improving maintainability.
The `server` crate now depends on `doctate-common`.

Cargo.lock

@@ -348,6 +348,7 @@ dependencies = [
  "axum",
  "axum-extra",
  "bcrypt",
+ "doctate-common",
  "dotenvy",
  "pulldown-cmark",
  "rand 0.8.6",