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`.

server/src/auth.rs

@@ -5,6 +5,7 @@ use std::time::Instant;
 use axum::extract::{FromRef, FromRequestParts};
 use axum::http::request::Parts;
 use axum_extra::extract::cookie::CookieJar;
+use doctate_common::API_KEY_HEADER;
 use tracing::warn;
 
 use crate::config::Config;
@@ -43,7 +44,7 @@ where
 
         let api_key = parts
             .headers
-            .get("X-API-Key")
+            .get(API_KEY_HEADER)
             .and_then(|v| v.to_str().ok())
             .ok_or(AppError::Unauthorized)?;