If you copy a song, using the Copy button, the new copy will be independent and can be edited without affecting the original. If you change the projects that a song is assigned to, using the Projects checkboxes, the same song will appear in both projects and any edits will affect both projects. It sounds like you either did the latter instead of the former, or you were still viewing the original songs in the original project when you removed the MIDI presets.
If you copy a song, you will need to re-upload recordings to the new copy; the recordings won't be copied automatically. The names can be the same. If you change the projects that a song is assigned to, the recordings will automatically be assigned to the same projects.