Merged PR 57851: internal/release/9.0 - merge from public

#### AI description  (iteration 2)
#### PR Classification
This pull request merges public branch changes into the internal 9.0 release, updating dependency versions, SDK settings, and build pipeline configurations.

#### PR Summary
The PR integrates upstream release changes by bumping key dependency and SDK versions and modernizing pipeline images.
- **`eng/Version.Details.xml`**: Upgraded dependency versions and SHA values for Arcade, Templating, and Helix SDK.
- **`global.json` and `eng/Versions.props`**: Incremented the .NET SDK version (from 9.0.112 to 9.0.113) and updated the release version prefix (from 9.0.13 to 9.0.14) along with the templating version.
- **Pipeline templates**: In files like **`eng/common/core-templates/post-build/post-build.yml`**, **`job/publish-build-assets.yml`**, and **`job/source-build.yml`**, updated image tags from `windows.vs2019.amd64` to `windows.vs2022.amd64` and adjusted Linux image settings.
<!-- GitOpsUserAgent=GitOps.Apps.Server.pullrequestcopilot -->

Author: vseanreesermsft

NuGet.config

@@ -5,6 +5,7 @@
     <!--Begin: Package sources managed by Dependency Flow automation. Do not edit the sources below.-->
     <!--  Begin: Package sources from dotnet-runtime -->
     <add key="darc-int-dotnet-runtime-1f0cdd4" value="https://pkgs.dev.azure.com/dnceng/internal/_packaging/darc-int-dotnet-runtime-1f0cdd4e/nuget/v3/index.json" />
+    <add key="darc-int-dotnet-runtime-9ecbfd4" value="https://pkgs.dev.azure.com/dnceng/internal/_packaging/darc-int-dotnet-runtime-9ecbfd4f/nuget/v3/index.json" />
     <!--  End: Package sources from dotnet-runtime -->
     <!--End: Package sources managed by Dependency Flow automation. Do not edit the sources above.-->
     <add key="dotnet-eng" value="https://pkgs.dev.azure.com/dnceng/public/_packaging/dotnet-eng/nuget/v3/index.json" />
@@ -22,6 +23,7 @@
     <!--Begin: Package sources managed by Dependency Flow automation. Do not edit the sources below.-->
     <!--  Begin: Package sources from dotnet-runtime -->
     <add key="darc-int-dotnet-runtime-1f0cdd4" value="true" />
+    <add key="darc-int-dotnet-runtime-9ecbfd4" value="true" />
     <!--  End: Package sources from dotnet-runtime -->
     <!--End: Package sources managed by Dependency Flow automation. Do not edit the sources above.-->
   </disabledPackageSources>