TeamCity CI - Make custom build output folder

I wish did fix the issue. I don't know whether TeamCity supports this, but this should be easy to script by yourself - as a last step of build in ProjectC add a script that will checkout ProjectC directories (not whole repository, just a directory where you keep binaries of ProjectC) from ProjectA and ProjectB, copy all required files and commit with some autogenerated message - seems quite simple.
However - this is dangerous. Build of ProjectA and ProjectB could break because an API of ProjectC changed, or some untested part of code was broken. This could disorganize work of teams A and B. I would go for solution where new version of libraries are commited once a day / a week / any other time period.

wish help you to fix your issue Preferred way is to use publishing targets in MSBuild.
Add new build step with runner type MSBuild Set Build file path to your web project csproj file Set Target to Clean;Build;Publish Set Command line parameters to /p:Configuration=Release;PublishDir=\\your\target\path

wish help you to fix your issue You should instead create an additional "dist" folder (or similar) and copy the exe and dll files to that folder after the build. There are many MSBuild properties and targets that can place files in the output folder, and attempting to change this behavior could break functionality.
Also, if you specify BaseIntermediateOutputPath then you should not specify IntermediateOutputPath. In the case of the properties above, you appear to be trying to use this:

<BaseIntermediateOutputPath>$(BUILD_INTERMEDIATE)\inter\</BaseIntermediateOutputPath>

this one helps. The following MSBuild script works, but I have to hard code the work folder (GetAssemblyIdentity line) that TeamCity uses. , You could use the built-in teamcity variables. Change your code to:

<Target Name="GetVersion">
<GetAssemblyIdentity AssemblyFiles="$(teamcity_build_checkoutDir)\AAA.Online.Web\bin\AAA.Online.Web.dll">
  <Output TaskParameter="Assemblies" ItemName="myAssemblyInfo"/>
</GetAssemblyIdentity>

<Message Text="Path is $(teamcity_build_checkoutDir)"/>

<PropertyGroup>
  <Pattern>(\d+)\.(\d+)</Pattern>
  <In>%(myAssemblyInfo.Version)</In>
  <OutVersion>$([System.Text.RegularExpressions.Regex]::Match($(In), $(Pattern)))</OutVersion>
</PropertyGroup>
</Target>

it fixes the issue You can't change the build output folder name with the current configuration options.
Moreover, you shouldn't. This is a part of the philosophy behind create-react-app: they say Convention over Configuration.