Concerning the Team Explorer :
Microsoft never open the customization of the Team Explorer by providing corresponding API in the Object Model. You're not supposed to add/modify/change nodes there. But people succeeded by doing some reflection tricks.
I still don't think it's a good idea to customize the Team Explorer, especially when you see Visual Studio 2012: they revamp everything there.
Microsoft Test Manager :
Looks like the data is pulled from the Team Build API directly, so you won't have the possibility to plug Team City there.
The Work Item field is a different story :
The values present are built from a Global List which is updated by Team Build. So you can maintain the list content instead of Team Build if you use Team City, Sean Lynch gave you the pointer to update a global list. This point is not hard at all, it's pretty straightforward.
All in all what you need the most if the Work Item field and you can do that. If you still need a list of all your Team City builds in Visual Studio, write a Visual Studio .vsix to add a custom View showing your Team City Build. I have less hope for MTM.
Last comment: stay away from the TFS Database, Microsoft is clear that you're not allowed to read/touch it. It's tempting, but you'll lose all MS support if you do it. But note it would be the only way to achieve what you want...