This ought to be pretty straightforward.
On ThirdConfig add two artifact dependencies. One whose source is CommonProject, and another whose source is WebProject. When configuring an artifact dependency it will allow you to specify which artifact files are are actually pulled from CommonProject and WebProject into ThirdConfig via the 'Artifact paths'. The artifact files can then be placed into some new folder hierarchy specific to ThirdConfig by using the 'Destination path'. These two options ought to be enough to create the directory structure that is the merging of CommonProject and WebProject. That takes care of the merge part.
The deploy is a bit more tricky. To my knowledge TeamCity does not support any sort of 'copy or upload to external location' function out of the box. For this bit you'll need to create an msbuild script (or batch file, or anything that can be run from the command line). Said script can expect the file/directory structure you've created via artifact dependencies where the root of the structure is the initial working directory of the script, and need only push these files out to your specific deploy location. That 'push' of course is going to be specific to your environment. Ftp, unc share, etc.