fix: stabilize TypeScript LSP startup

Seed new TypeScript LSP profiles with --stdio, validate that shared profiles keep the required flag, auto-add it for existing profiles in the sidecar runtime, and include recent stderr when a language server dies before completing a request. Add shared and sidecar regression coverage.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

sidecar/tests/Eryx.AgentHost.Tests/LspToolSessionTests.cs

@@ -1,4 +1,5 @@
 using System.Text.Json;
+using Eryx.AgentHost.Contracts;
 using Eryx.AgentHost.Services;
 
 namespace Eryx.AgentHost.Tests;
@@ -24,4 +25,32 @@ public sealed class LspToolSessionTests
 
         Assert.Contains("relativePath", json);
     }
+
+    [Fact]
+    public void ResolveProcessArguments_AddsStdioForTypeScriptLanguageServer()
+    {
+        RunTurnLspProfileConfigDto profile = new()
+        {
+            Command = "typescript-language-server",
+            Args = [],
+        };
+
+        IReadOnlyList<string> args = LspToolSession.ResolveProcessArguments(profile);
+
+        Assert.Equal(["--stdio"], args);
+    }
+
+    [Fact]
+    public void ResolveProcessArguments_DoesNotDuplicateStdioWhenAlreadyPresent()
+    {
+        RunTurnLspProfileConfigDto profile = new()
+        {
+            Command = "typescript-language-server",
+            Args = ["--stdio"],
+        };
+
+        IReadOnlyList<string> args = LspToolSession.ResolveProcessArguments(profile);
+
+        Assert.Equal(["--stdio"], args);
+    }
 }