[M0-S3] CI: typecheck and build the game on every PR #25

Problem

Docs and art are checked in CI already; the game isn't. A type error should fail the PR, not the demo.

Acceptance criteria

• A game workflow runs npm ci, npm run typecheck, npm test and npm run build in game/ on PRs that touch it
• The existing Documents and Art pipeline workflows keep running

Notes and dependencies

Owner: Vatsal – lane B, game code (docs/13).

Verification

A PR with a deliberate type error fails the check; the fix makes it green.