How do I move the debug menu to the footer or problems view in Visual Studio Code?

There are a few issues about this on vscode github, for example see summary of issues

There is an extension available to move the debug menu bar to the status bar which looks good:

StatusBar Debugger

----- Update -----------

In v1.42 and 1.43 it is now easy to move various debug view to the panel. See https://github.com/microsoft/vscode-docs/blob/vnext/release-notes/v1_43.md#easier-moving-of-even-more-views and VS Code - toggle search icon in Activity Bar, move from panel or back and especially

Visual Studio Code: How to have both Debug window and Explorer window open together?

Easier moving of even more views

Last release, we had just a taste of relocating views to the panel. In this relase, this experience has been expanded to many more views. You can now move many more views such as the debug views or the new timeline view. You can also now move them into other viewlets in the sidebar. For example, perhaps you'd like to move the Open Editors view into the SCM viewlet. You can do all of this by dragging the views and dropping them in the desired location in the Activity Bar or Panel. The gif below highlights a few examples.

We've introduced a new command to make moving views easier with the keyboard: View: Move Focused View (workbench.action.moveFocusedView).

And finally, this is a preview feature. So, in case you get into a state that you can't fix, there is a command to reset all views to their original locations: View: Reset View Locations (workbench.action.resetViewLocations).

It looks like you can dock or hide the debug tool bar:

To hide the debug tool bar change the "debug.toolBarLocation" setting to hidden.

Either floating in all views, docked in the debug view, or hidden.

Docking is better than floating and hidden. Adding the extension mentioned in the other answer is also helpful.