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

[stooit]

Summary

Fixes all failing tests and type errors in the Bun + Hono monorepo. All 22 tests pass and tsc --noEmit is clean.

Changes

• shared: implement paginate<T>() — was a stub throwing "not implemented". Now slices by (page-1)*size, computes total/totalPages, and returns { data, page, pageSize, total, totalPages }. Handles empty arrays, partial last pages, and out-of-range pages.
• shared: rename User.userName → username — the field name was inconsistent between packages; aligned to what the (unmodifiable) tests expect, updating all handler/db usages.
• api: add missing badRequest import in routes/users.ts (was "Cannot find name 'badRequest'").
• api: fix auth middleware HTTP-method case bug — public-method comparison used lowercase 'post' where Hono's c.req.method returns uppercase 'POST', so POST /users was incorrectly requiring a token.
• tsconfig: scope types to ["bun-types"] — resolves bun:test / process type errors under tsc. bun-types is an existing devDependency; no new dependencies added.

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → no errors

Assumptions

• The tests are the source of truth for the correct username field name.
• Per the "fix only what the tests require" constraint, non-blocking review suggestions (a size <= 0 guard in paginate, echoing out-of-range page, a restored policy comment) were intentionally left out of scope — no test exercises those paths and no current caller invokes paginate.

Constraints honoured

• No test files modified
• No new dependencies added