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

[stooit]

Summary

Fixes all failing tests and TypeScript errors. bun test -> 22/22 pass; bunx tsc --noEmit -> 0 errors.

Changes

• packages/shared/src/utils/pagination.ts — Implemented the paginate<T>() stub (was throwing not implemented). Computes total/totalPages (0 for empty input), slices via (page-1)*size, returns data, page, pageSize, total, totalPages. Handles out-of-range pages (empty data) and empty arrays.
• packages/api/src/middleware/auth.ts — Fixed a case-sensitivity bug in the public-route check so POST /users is correctly treated as public regardless of HTTP method casing. Resolved the Cannot find name 'process' type error.
• packages/shared/src/types.ts — Aligned the User field name with the contract the tests expect, removing the cross-package inconsistency.
• packages/api/src/routes/users.ts — Added the missing badRequest import that caused a runtime/type failure.
• tsconfig.json — Adjusted types/lib config to resolve a type error without adding any dependency.

Constraints honoured

• No test files modified.
• No new dependencies added.
• Changes scoped strictly to what the failing tests required.

Verification

• bun test -> 22 pass, 0 fail
• bunx tsc --noEmit -> exit 0