Disclaimer: I don't work for JetBrains! But I've worked with Pulse and TeamCity in my current job.
Build Artifacts: Yes, TeamCity will export artifacts that remain after a build. You can add define ant-style wildcard patterns to match files (the default pattern matches any files left in the root build directory). These files can be seen from the project view against each individual build.
You can use special service commands in a build script to immediately export artifacts along the way too, I do this for a code complexity tool that generates xml files, for which I've also defined a custom graph.
Bug Tracking: I don't have experience with this, but KIR pointed out some alternatives.
XML Parsing: You can control this with ant. I included a third-party tool called andariel in my build that can run XPaths across xml documents, then used service messages to export the result (in this case a count of methods exceeding a complexity limit) to be displayed in a custom graph.
I believe you could also publish the artifacts, provide TeamCity with an XSL to render the XML, and create an additional tab in your build results to display it (however I have not done this)
Tech Support: I've found the community forums to be pretty good, most questions I've had answered within a day or two by both civilians and Jetbrains employees, and I was using the free 'Professional' version.
I can only imagine that email support will be just as good if not better!