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

[stooit]

Summary

Fixes all 9 failing tests and all 6 type errors in the monorepo. Result: 22 pass / 0 fail, npx tsc --noEmit clean.

Changes

• packages/shared/src/utils/pagination.ts — implemented paginate() per the test contract (was a throw "not implemented" stub): computes page slice, total, totalPages, and echoes page/pageSize.
• packages/api/src/routes/users.ts — added the missing badRequest import (was a runtime ReferenceError, breaking the POST /users 400-on-missing-fields test).
• packages/shared/src/types.ts — renamed User.userName → username to match the field the tests (authoritative) and route handlers expect, resolving the cross-package mismatch.
• packages/api/src/middleware/auth.ts — fixed the HTTP-method case-sensitivity bug so POST /users is correctly treated as a public route (no token required).
• tsconfig.json — added "types": ["bun-types"] so process and bun:test resolve during typecheck. No new dependency (bun-types was already a devDependency).

Verification

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

Constraints honoured

• No test files modified.
• No new dependencies added.
• Scope limited to what the tests require.

Assumptions

• username (lowercase) is canonical — chosen because the (unmodifiable) tests use it.
• Reviewer noted optional, non-blocking hardening for paginate edge inputs (negative/zero page/size) and general auth robustness. These are out of scope for "fix only what the tests require" and were intentionally not changed.

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