fix: resolve all failing tests and type errors

[stooit]

Summary

Fixes all failing tests and TypeScript errors across the api and shared packages. Final state: 22 tests pass, 0 fail; tsc --noEmit clean.

Changes

• packages/shared/src/utils/pagination.ts — implemented paginate() (was a stub throwing not implemented). Handles page slicing, total/totalPages, page/pageSize, out-of-range pages, empty arrays, and partial last pages per the test contract.
• packages/api/src/middleware/auth.ts — fixed HTTP method case-sensitivity bug in the public-route check ("post" → "POST"), so POST /users is correctly treated as public. Also resolved the process global type error.
• packages/api/src/routes/users.ts — added the missing badRequest import and reconciled the user field name.
• Shared User type / field naming — reconciled the inconsistent field name to username to match the contract exercised by the (unmodified) tests.
• tsconfig.json — added bun-types to compilerOptions.types to resolve process and bun:test type-resolution errors. No new dependencies added (bun-types already present).

Verification

• bun test → 22 pass / 0 fail
• bunx tsc --noEmit → clean

Constraints honoured

• No test files modified. No new dependencies added. Changes minimal and scoped to what tests/types require.

Assumptions

• Where the User field name was inconsistent (userName vs username), the unmodifiable tests are the source of truth, so non-test code was aligned to username.