You said it was a maven web app. So when you imported it into STS did you use file->import->Maven->Existing Maven Projects ?
You should use that wizard and then browse to the projects root POM file.
Once you do this you should be able to right click the project choose Run as -> Maven install That will build the project. After that you should see your war file in the target directory in your package explorer.
Also in STS there is no need to install Tomcat. The TC server which comes pre-configured is basically a glorified Tomcat 7. Right clicking your project and selecting Run as -> Run on Server should do the trick. Alternatively double click VMware Vfabric tc Server... in the bottom left server tab. There are three tabs along the bottom of the widow that opens; overview, insight , and modules. Select modules and press the Add Web Module button. If your project is a web app it should show up there and you can add it. Remember to press Ctrl-S or the save button on the top tool bar to save any changes you make.