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

[stooit]

Summary

Fixes all failing tests and eliminates all TypeScript type errors in the monorepo. After these changes bun test reports 22 pass / 0 fail and npx tsc --noEmit is clean (exit 0).

Changes

• Auth middleware (packages/api/src/middleware/auth.ts): fixed a case-sensitivity bug — the public-methods allow-list used lowercase 'post' but Hono's c.req.method is uppercase, so POST /users was incorrectly requiring a token. Also fixes the process type error via the tsconfig change below.
• Shared types (packages/shared/src/types.ts): renamed the User field userName → username to match the canonical name used by the tests and the API routes (cross-package inconsistency).
• Users route (packages/api/src/routes/users.ts): added the missing badRequest import from ../lib/errors (was used but not imported, causing a runtime + type failure). Removed a now-stale // BUG: comment.
• Pagination utility (packages/shared/src/utils/pagination.ts): implemented the previously-stubbed paginate() per the test contract — computes total, totalPages (0 when empty), slices the correct page, and returns { data, page, pageSize, total, totalPages }. Out-of-range pages yield an empty data array.
• tsconfig (tsconfig.json): added typeRoots + types (node, bun-types) so bun:test and process resolve. Packages were already installed; no new dependencies added.

Verification

• bun test → 22 pass, 0 fail (37 assertions, 4 files)
• npx tsc --noEmit → no errors

Assumptions

• Test files are canonical and were not modified; source was made to conform to the tests.
• username (not userName) is the canonical field name, as used by the tests.
• paginate() does not clamp negative/zero page sizes or negative pages — these cases are outside the test contract and the task scope ("fix only what the tests require"). A follow-up could add input guards if desired.