fix: repair failing tests and type errors across api and shared packages

[stooit]

Summary

Fixes all failing tests and TypeScript errors across the api and shared packages. Final state: 22/22 tests pass and tsc --noEmit is clean.

Changes

• Pagination (packages/shared/src/utils/pagination.ts) — implemented the paginate<T>() utility, which was a throwing stub. Computes total and totalPages, slices by (page-1)*size, and returns empty data for out-of-range pages.
• Auth middleware (packages/api/src/middleware/auth.ts) — fixed an HTTP-method case-sensitivity bug so POST /users is correctly treated as a public (no-token) route.
• User field naming (packages/shared/src/types.ts, packages/api/src/routes/users.ts) — aligned the User type field name with the spelling the tests expect, resolving cross-package type mismatches, and imported the missing badRequest response helper in the users route.
• tsconfig.json — included the already-installed node/bun type definitions to resolve process / bun:test type-resolution errors. No new dependencies were added.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → clean (exit 0)

Constraints honoured

• No test files modified
• No new dependencies added
• Fixed only what the tests required

Assumptions

• The test files are the source of truth for the canonical User field name; source types and route handlers were aligned to match them.
• The required node/bun type packages were already present in node_modules, so they were referenced via tsconfig rather than installed.