So my situation is that I finally finished configuring TeamCity for CI. I got it to run my unit tests with some friendly help on SO.
However, many unit tests fail be
Your problem is not to do with TeamCity I don't think, it's to do with the way that MSTest works. You need your .config file to be a DeploymentItem and have your tests deploy it to the directory that MSTest will run the tests in.
To be honest I'm surprised that you don't have this problem running locally, and it makes me think that you must be using some other test runner (like ReSharper) to run the tests if you have not seen this problem on your local machines.