fix: resolve failing tests across api and shared packages

[stooit]

Summary

Fixes all failing tests and the code-level type errors in the monorepo. 22/22 tests pass. No test files modified, no dependencies added.

Fixes

• api/src/routes/users.ts — badRequest was called but never imported, causing a ReferenceError (500 instead of 400 on missing fields). Added the missing import.
• api/src/middleware/auth.ts — public-methods list contained lowercase "post", which never matched Hono's uppercase c.req.method, so POST /users wrongly required a token (401 instead of 201). Corrected to "POST".
• shared/src/utils/pagination.ts — implemented the paginate<T>() stub: correct page slicing, partial last page, total/totalPages, page/pageSize fields, and empty/out-of-range handling.
• shared/src/types.ts — reconciled inconsistent field name across the shared and api packages.

Verification

• bun test → 22 pass / 0 fail.
• tsc --noEmit: errors reduced from 33 (on main) → 8. The remaining 8 are pre-existing environment-config issues (bun:test module and process global untyped — no bun-types/@types/node in compilerOptions.types). Resolving them would require adding dependencies, which the task explicitly forbids. None relate to the failing tests.

Assumptions

• "Fix only what the tests require" + "do not add dependencies" → the pre-existing untyped-process/bun:test tsc errors were left as-is.
• Independent verification confirmed all 8 remaining tsc errors also exist on main (33 baseline errors), so none were introduced by this change.

Review

Independently reviewed (review subagent): APPROVE — auth fix introduces no bypass risk, pagination edge cases correct, type reconciliation consistent across packages.