feat: integrate Datadog APM + Sentry error tracking with CloudWatch metrics FRE-4806

- Add CloudWatch metrics emitter (api_latency, api_requests, api_errors)
- Add request monitoring middleware for API (latency, error rate, throughput)
- Register error-handling, logging, and monitoring middleware in server.ts
- Add Datadog log forwarding via HTTP intake API
- Add application-level CloudWatch alarms for P99 latency, error rate, throughput
- Inject Datadog/Sentry env vars and secrets into ECS task definitions
- Add DD_API_KEY and SENTRY_DSN to ECS secrets
- Create CloudWatch log groups for datadog and sentry services
- Update .env.example with AWS_REGION and monitoring variables
- Add @aws-sdk/client-cloudwatch dependency to monitoring package

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

infra/modules/ecs/main.tf

@@ -96,6 +96,50 @@ resource "aws_ecs_task_definition" "services" {
         {
           name  = "PORT"
           value = tostring(each.port)
+        },
+        {
+          name  = "DD_ENV"
+          value = var.environment
+        },
+        {
+          name  = "DD_SERVICE"
+          value = "${var.cluster_name}-${each.key}"
+        },
+        {
+          name  = "DD_VERSION"
+          value = var.container_images[each.key]
+        },
+        {
+          name  = "DD_TRACE_ENABLED"
+          value = "true"
+        },
+        {
+          name  = "DD_LOGS_INJECTION"
+          value = "true"
+        },
+        {
+          name  = "DD_AGENT_HOST"
+          value = "localhost"
+        },
+        {
+          name  = "DD_AGENT_PORT"
+          value = "8126"
+        },
+        {
+          name  = "SENTRY_ENVIRONMENT"
+          value = var.environment
+        },
+        {
+          name  = "SENTRY_RELEASE"
+          value = var.container_images[each.key]
+        },
+        {
+          name  = "AWS_REGION"
+          value = "us-east-1"
+        },
+        {
+          name  = "DD_SITE"
+          value = "datadoghq.com"
         }
       ]
 
@@ -115,6 +159,14 @@ resource "aws_ecs_task_definition" "services" {
         {
           name      = "RESEND_API_KEY"
           valueFrom = "${var.secrets_arn}:RESEND_API_KEY::"
+        },
+        {
+          name      = "SENTRY_DSN"
+          valueFrom = "${var.secrets_arn}:SENTRY_DSN::"
+        },
+        {
+          name      = "DD_API_KEY"
+          valueFrom = "${var.secrets_arn}:DD_API_KEY::"
         }
       ]