fix: resolve all failing tests across api and shared packages

[stooit]

Summary

• Auth middleware case-sensitivity: The publicMethods array used lowercase "post" but HTTP methods from Hono arrive as uppercase "POST". Fixed to match correctly.
• Missing import in users route: badRequest was used but not imported from ../lib/errors. Added the import.
• User type field name mismatch: The User type in packages/shared/src/types.ts used userName but all tests and route handlers expect username. Renamed the field.
• Pagination utility stub: paginate() in packages/shared/src/utils/pagination.ts was throwing "not implemented". Implemented it to return { data, page, pageSize, total, totalPages }.
• TypeScript process error: Added a declare var process type declaration in auth.ts to resolve the TS2580 error without adding @types/node.

Verification

• All 22 tests pass (bun test)
• Zero TypeScript errors in source files (npx tsc --noEmit — remaining errors are only in test files due to missing bun:test type declarations)

Assumptions

• Test files are the source of truth and were not modified
• No new dependencies were added
• The process declaration is minimal and scoped to the single file that uses it