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 tests pass, 0 fail; tsc --noEmit clean.

Changes

• shared/src/utils/pagination.ts — implemented the paginate() stub (previously threw "not implemented"). Handles 1-indexed page slicing, total, totalPages rounding, out-of-range pages (→ empty data), and empty input.
• shared/src/types.ts — renamed User.userName → username to align the shared type with the route handlers and the (unmodifiable) test contracts.
• api/src/middleware/auth.ts — fixed a case-sensitivity bug in the public-method allow-list so POST /users is correctly treated as public (no token required); also resolved the Cannot find name 'process' type error.
• api/src/routes/users.ts — added the missing badRequest import causing the "returns 400 for missing fields" handler to fail at runtime.
• tsconfig.json — wired compiler types so bun:test and process resolve (no new dependencies added).

Verification

• bun test → 22 pass / 0 fail
• bunx tsc --noEmit → exit 0
• Independent review pass: pagination edge cases, auth casing fix, and the username rename confirmed consistent across both packages.

Assumptions

• The (unmodifiable) tests are the source of truth for field names and contracts; source was aligned to them (username, not userName).
• No test files were modified and no new dependencies were added, per task constraints.

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