Add structured logging with log/slog

Replace fmt.Printf logging calls with slog.Info/slog.Error for structured
JSON output to stderr. Add internal/log package to initialize the default
slog handler from the config log level. Fix .gitignore to only ignore the
binary at the repo root, not the cmd/eng-pad-server directory.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

.gitignore

@@ -1,4 +1,4 @@
-eng-pad-server
+/eng-pad-server
 /srv/
 *.db
 *.db-wal