fix: make test suite pass and eliminate type errors

[stooit]

Summary

Fixes all failing tests (now 22 pass / 0 fail) and eliminates all tsc --noEmit type errors across the api and shared packages.

Changes

• Pagination (packages/shared/src/utils/pagination.ts): implemented the paginate<T> stub — page slice, total, totalPages, page/pageSize; handles partial last pages, out-of-range pages, and empty arrays.
• Shared type consistency (packages/shared/src/types.ts): renamed User field userName → username to match every consumer (db, route, tests).
• Auth middleware (packages/api/src/middleware/auth.ts): fixed a case-sensitivity bug — the public-method allow-list checked "post" against Hono's uppercase c.req.method, so POST /users was wrongly treated as protected.
• Users route (packages/api/src/routes/users.ts): added the missing badRequest import that broke the missing-fields validation path.
• tsconfig: added bun-types to compilerOptions.types to resolve process / bun:test global type errors (no new dependencies).

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → clean

Assumptions / notes

• Followed "fix only what tests require; do not modify tests; no new dependencies". Tests were authoritative for the username spelling.
• Review flagged a non-blocking edge case: paginate does not guard degenerate inputs (size<=0/page<=0). Not exercised by the suite; adding a throw/clamp would introduce unspecified behaviour, so left out per the mandate. Worth a follow-up if pagination query params get exposed without validation.