feat(observability): integrate Sentry error tracking for API and Web apps

- API: add @sentry/nestjs with instrument.ts, SentryModule, and SentryGlobalFilter
- Web: add @sentry/nextjs with client/server/edge configs, instrumentation hook
- Update next.config.js with withSentryConfig wrapper
- Replace TODO in error.tsx with Sentry.captureException
- Add SENTRY_DSN, SENTRY_AUTH_TOKEN, SENTRY_ORG, SENTRY_PROJECT to .env.example

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/api/src/instrument.ts

@@ -0,0 +1,11 @@
+import * as Sentry from '@sentry/nestjs';
+import { nodeProfilingIntegration } from '@sentry/profiling-node';
+
+Sentry.init({
+  dsn: process.env['SENTRY_DSN'],
+  environment: process.env['NODE_ENV'] ?? 'development',
+  integrations: [nodeProfilingIntegration()],
+  tracesSampleRate: process.env['NODE_ENV'] === 'production' ? 0.2 : 1.0,
+  profilesSampleRate: process.env['NODE_ENV'] === 'production' ? 0.2 : 1.0,
+  enabled: !!process.env['SENTRY_DSN'],
+});