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

[stooit]

Summary

Fixes all failing tests and eliminates all tsc --noEmit type errors in the api (Hono) and shared packages. Final state: 22 pass / 0 fail, tsc clean.

Changes

• packages/shared/src/utils/pagination.ts — Implemented the paginate<T>() stub. Correctly handles partial last page, out-of-range pages (returns empty data), empty arrays, and totalPages calculation (with a zero-size guard).
• packages/api/src/middleware/auth.ts — Fixed a case-sensitivity bug: the method allow-list used lowercase ('post') while c.req.method returns uppercase ("POST"), so valid POSTs were rejected with 401. Normalised the allow-list to uppercase. No auth bypass introduced.
• packages/api/src/routes/users.ts — Added the missing badRequest import; aligned the create payload to use username.
• packages/shared/src/types.ts — Made the User.username field name consistent across both packages (tests expect username).
• tsconfig.json — Added "types": ["bun-types"] so bun:test and the process global resolve under tsc. No new dependencies added.

Verification

• bun test → 22 pass, 0 fail (4 files)
• bunx tsc --noEmit → clean
• Independent review pass confirmed correctness with no blocking issues.

Constraints honoured

• No test files modified.
• No dependencies added.
• Only the minimal changes required by the tests.

Assumptions

• The username/userName inconsistency resolved toward username because the (unmodifiable) tests use username.
• The bun:test / process errors were type-resolution config, not missing packages — resolved via tsconfig types.