Add a DocFX documentation gate for the API

The backend workflow now builds the API reference with DocFX and treats
every warning as an error, so an unresolved reference or a broken link
fails the check.

- docs/api holds the DocFX project: it reads apps/api/Progmaweb.Api.csproj
  and includes internal members, because the service is an application and
  most of its types are internal.
- .config/dotnet-tools.json pins docfx 2.81.0, so CI and a local run use the
  same version through `dotnet tool restore`.
- The workflow also runs when the DocFX project or the tool manifest change.
- `just api-docs` runs the same two commands locally.
- The generated site and metadata are ignored.

The gate does not require documentation comments: the API has none yet, and
DocFX does not report an undocumented member. Requiring them is a separate
change to the C# sources.

Verified locally with docfx 2.81.0 on the .NET 10 SDK: the build passes with
no warnings, and a broken link in the overview page makes it fail.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

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