As said you have to tell cp what to copy and where to put it.
cp -r is recursive so if you give it a directory as the source it copies that directory and all its content. if you give it a directory as the destination it puts what it is copying into that directory, thus your extra level of directories. You copied a directory into another directory of the same name.
You could have used 2 different ways to do that.
Code:
cd # first to make certain where you are.
cp -ar Documents/* /mnt/backup/Documents/
or
cp -ar Documents /mnt/backup/
Either way would have done exactly what you expected.
Read the man page for cp to see what the -a option does. Probably what you want.
For your second question.
I would suggest you dismount that directory/drive before you start an install. It can always be mounted and accessed after the install is complete. Leaving it mounted will (usually) not be an issue since rebooting to start the install will dismount it, but better to do it manually than to rely on the system doing it for you.