feat: wire Sentry, conditional on DSN env var

apps/web (@sentry/nextjs):
- sentry.client.config.ts / sentry.server.config.ts / sentry.edge.config.ts
  initialize Sentry only when NEXT_PUBLIC_SENTRY_DSN (or SENTRY_DSN on the
  server) is set; tracesSampleRate 0.1
- next.config.mjs wraps the existing config with withSentryConfig only when a
  DSN is present; source-map upload stays disabled until SENTRY_AUTH_TOKEN is
  provided
- .env.example documents NEXT_PUBLIC_SENTRY_DSN and the optional auth token

apps/api (@sentry/node):
- src/lib/sentry.ts initializes Sentry at module load when SENTRY_DSN is set
- server.ts imports sentry.ts as the very first side-effect so early-boot
  errors (env validation, plugin registration) reach Sentry
- env schema gains optional SENTRY_DSN (URL)
- app.ts registers an errorHandler that captures the exception with the
  authenticated user when SENTRY_DSN is set; logs and re-sends as before

Build-time housekeeping:
- profile route: targetDate ?? null on create to satisfy Prisma's input shape
  under exactOptionalPropertyTypes
- profile form: useForm receives defaultValues only when initial is provided
  (spread instead of `defaultValues: undefined`); Field error prop typed
  `string | undefined` for exactOptionalPropertyTypes

Without a DSN both apps run unchanged (Sentry is inert).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

apps/web/next.config.mjs

@@ -1,3 +1,4 @@
+import { withSentryConfig } from '@sentry/nextjs';
 import createNextIntlPlugin from 'next-intl/plugin';
 
 const withNextIntl = createNextIntlPlugin('./src/i18n.ts');
@@ -20,4 +21,17 @@ const nextConfig = {
   },
 };
 
-export default withNextIntl(nextConfig);
+const isSentryEnabled = Boolean(process.env.NEXT_PUBLIC_SENTRY_DSN);
+
+const sentryWebpackPluginOptions = {
+  silent: true,
+  org: process.env.SENTRY_ORG,
+  project: process.env.SENTRY_PROJECT,
+  // Source map upload requires SENTRY_AUTH_TOKEN — disabled until configured.
+  disableServerWebpackPlugin: !process.env.SENTRY_AUTH_TOKEN,
+  disableClientWebpackPlugin: !process.env.SENTRY_AUTH_TOKEN,
+};
+
+const composed = withNextIntl(nextConfig);
+
+export default isSentryEnabled ? withSentryConfig(composed, sentryWebpackPluginOptions) : composed;