fix: resolve all failing tests across api and shared packages

[stooit]

Summary

• Auth middleware case-sensitivity: Fixed HTTP method comparison in publicMethods array — changed "post" to "POST" since Hono provides uppercase methods per RFC 7231
• Missing import: Added badRequest to the import from ../lib/errors in users route handler
• Field name inconsistency: Renamed userName → username in the shared User type to match test expectations
• Pagination utility: Implemented the paginate function (was a stub throwing "not implemented") — returns { data, page, pageSize, total, totalPages }
• TypeScript config: Added skipLibCheck: true to resolve process type errors without adding new dependencies

Verification

All 22 tests pass (bun test). The only remaining tsc --noEmit errors are bun:test module declarations in test files (Bun built-in, not resolvable without modifying test files or adding dependencies).

Assumptions

• Test files are the source of truth and must not be modified
• No new dependencies may be added
• The bun:test TypeScript errors in test files are acceptable (Bun runtime resolves them)