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

[stooit]

Summary

Fixes all failing tests (now 22/22 pass) and eliminates all tsc --noEmit type errors across the api and shared packages. Changes are minimal and confined to source/config — no test files modified, no dependencies added.

Changes

• packages/shared/src/utils/pagination.ts — Implemented the paginate() utility (was a throwing stub). Handles empty input, out-of-range pages, partial last pages, and computes total/totalPages/page/pageSize correctly.
• packages/api/src/middleware/auth.ts — Fixed a case-sensitivity bug in the HTTP method comparison so POST /users is correctly treated as a public route. Also resolved the process type error.
• packages/api/src/routes/users.ts — Added the missing badRequest import that caused a runtime failure on the missing-fields path.
• packages/shared/src/types.ts — Renamed User.userName → username to resolve a field-name inconsistency between the shared type and its consumers/tests.
• tsconfig.json — Added bun-types to the types array (already an installed devDependency — not a new dependency).

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → exit 0, no errors
• Independent code review confirmed correctness, consumer consistency of the rename, and that no test files or dependencies were touched.

Assumptions

• The test files are the source of truth for the canonical username field name; the shared type (userName) was the inconsistent side and was aligned to the tests rather than editing tests.
• bun-types was treated as a config fix (existing devDependency) rather than a new dependency.