feat: add GitHub Actions CI and support hosting under a subpath

Mirrors the GitLab pipeline — typecheck, tests, feed sanity — and adds a
Pages deploy.

Pages serves from /<repo>/, and the app used absolute asset paths, so that
job would have shipped a site that 404s on everything. Assets now resolve
against a <base href> the build substitutes, the feed URL resolves against
document.baseURI so deep links work too, and the service worker derives its
paths from its own registration scope. Root-hosted builds are unchanged.

Co-Authored-By: Claude Opus 5 (1M context) <[email protected]>

src/client/manifest.webmanifest

@@ -2,15 +2,15 @@
   "name": "Event Clock",
   "short_name": "Event Clock",
   "description": "Live and upcoming gacha events, sorted by what expires next.",
-  "start_url": "/",
-  "scope": "/",
+  "start_url": "./",
+  "scope": "./",
   "display": "standalone",
   "background_color": "#12141c",
   "theme_color": "#12141c",
   "orientation": "portrait-primary",
   "icons": [
     {
-      "src": "/icon.svg",
+      "src": "icon.svg",
       "sizes": "any",
       "type": "image/svg+xml",
       "purpose": "any maskable"