Is there any way to cut a track from one project and paste it into another?

You could export the track, and then import it into the new project, or use the browser, and double click the wave at the timeline. 

Or open both projects at the same time and copy/cut/paste between them. To open more than one project at time make sure "Allow Only One Open Project at a Time" is unchecked in Preferences > File > Advanced. Select Advanced radio button at the bottom of the Preferences dialog to be able to access the Advanced file settings.

You mean, the track strip with all its settings, right?
I think the only way to do this is to save the selected track as a "track template", then insert it into the new project, then copy the track events with normal cut-n-paste.
 
I, too, would enjoy the ability to copy a complete track into another project.