feat(installer): harden database init and service setup steps

- InitDatabaseStep: create DbPath parent directory so custom paths work
- RegisterServiceStep: pass obj= argument so ServiceAccount is honoured
- StartServiceStep: poll for RUNNING state so downstream steps don't race

src/ClaudeDo.Installer/Steps/StartServiceStep.cs

@@ -13,15 +13,26 @@ public sealed class StartServiceStep : IInstallStep
         progress.Report($"Starting {ServiceName}...");
 
         var (exit, _) = await ProcessRunner.RunAsync("sc.exe", $"start {ServiceName}", null, progress, ct);
-        if (exit == 0) return StepResult.Ok();
+        // 1056 = ERROR_SERVICE_ALREADY_RUNNING — fine, fall through to the readiness poll.
+        if (exit != 0 && exit != 1056)
+            return StepResult.Fail($"sc.exe start failed with exit code {exit}");
 
-        // Exit 1056 = ERROR_SERVICE_ALREADY_RUNNING — that's fine too.
         if (exit == 1056)
-        {
             progress.Report("Service was already running.");
-            return StepResult.Ok();
+
+        // sc.exe start returns as soon as SCM accepts the command. Poll until the
+        // service actually reports RUNNING so downstream steps and SignalR clients
+        // don't race the worker's startup.
+        progress.Report("Waiting for service to reach RUNNING state...");
+        for (var i = 0; i < 30; i++)
+        {
+            ct.ThrowIfCancellationRequested();
+            var (q, output) = await ProcessRunner.RunAsync("sc.exe", $"query {ServiceName}", null, progress, ct);
+            if (q == 0 && output.Contains("RUNNING", StringComparison.OrdinalIgnoreCase))
+                return StepResult.Ok();
+            await Task.Delay(1000, ct);
         }
 
-        return StepResult.Fail($"sc.exe start failed with exit code {exit}");
+        return StepResult.Fail("Service did not reach RUNNING state within 30 seconds.");
     }
 }