## Why

The current instance management has significant gaps in health monitoring. When starting instances, there's no verification that containers actually boot successfully - failures only surface when users try to access broken tunnels. The existing health check only validates tunnel URLs, not container health, leading to false positives where a "healthy" tunnel serves 502 errors from a crashed tool. Additionally, readiness probes exist as unused infrastructure, and auto-recovery blindly recreates tunnels on any HTTP error including legitimate 502s from the application itself.

## What Changes

- **Startup health checks**: Verify containers reach a running state after `docker compose up`, with clear failure messages when containers crash or fail to start
- **Container health checks**: Check container status via Docker API (`docker ps`, `docker inspect`) in addition to tunnel URL checks
- **Readiness probe integration**: Wire the existing `execute_probe()` service into the instance startup flow, using tool type configured probes
- **Smart auto-recovery**: Only recreate tunnels when the tunnel endpoint itself is unreachable (connection refused, timeout, DNS failure), NOT when the tool returns 502/503/504 errors
- **Instance status granularity**: Distinguish between "starting" (container booting), "running" (healthy), "unhealthy" (container up but probe failing), and "error" (failed to start)

## Capabilities

### New Capabilities
- `instance-startup-health`: Container startup verification and failure detection
- `instance-runtime-health`: Continuous health monitoring combining container and tunnel checks
- `readiness-probe-integration`: Tool-type configured readiness probes during instance startup
- `smart-tunnel-recovery`: Context-aware tunnel recreation that distinguishes tunnel failures from application errors

### Modified Capabilities
- `session-management-fixes`: Update health check endpoint to include container status, modify tunnel health logic to be smarter about error codes

## Impact

- **Backend**: `api/tool_instances.py` (start_instance, health check, recreate tunnel), `services/docker.py` (container status checks), `services/readiness_probe.py` (integration into startup flow)
- **Frontend**: `pages/sessions.tsx` (display new status states, show startup errors, smarter health badges)
- **Database**: No schema changes - uses existing `status` field with new state values
- **API**: New response fields in health check endpoint (container_status, probe_result, last_probe_at)