Enable type-aware Oxlint rules and compiler diagnostics

[skovhus]

Enable type-aware correctness rules and TypeScript compiler diagnostics in the shared Oxlint configuration, backed by pinned oxlint-tsgolint. Lint commands now check types, so remove the standalone typecheck script and run Vitest directly from the test commands. Verification runs Oxlint once before the build and tests. Build and watch commands retain tsc for JavaScript and declaration output.

Reference tsconfig.test.json from the root project so Oxlint discovers tests, helpers, and the Vitest config. Keep its Bundler module resolution separate from the production Node16 configs. Fix newly reported promise and string-conversion issues, preserve detached background work, and disable unbound-method only in test files where Vitest inspects mock methods.

Validation on the branch rebased onto main:

• pnpm install --frozen-lockfile
• pnpm verify:bail: lint, formatting, both builds, and coverage passed; 526 tests passed and 2 skipped.
• Temporary probes confirmed type mismatches, implicit any, and floating promises are rejected in server/client source, server/client tests, nested tests, test helpers, and the Vitest config. All probes were removed.
• Independent code review completed with no remaining findings.

Local lint/type-check benchmark (macOS arm64, Node 22.22.2, TypeScript 6.0.3): one warmup per mode and five alternating measured runs on the same checkout. The old Oxlint config plus tsc --noEmit -p tsconfig.test.json took a median 1.516s (tsc alone: 1.431s); the new type-aware Oxlint command took 0.577s. This saves about 0.94s / 62% for this step while adding type-aware lint rules. Measurements use the local executables, excluding pnpm startup, Prettier, builds, and tests. tsc still handles builds; coverage remains unchanged in this PR.