You could do a copy from the command line, or sudo nautilus and then do the drag and drop with that.
It sounds like you are trying to drag into a folder that your current login is not elevated enough for, you could also sudo chown <username> <path> if you want and it wont then give you the problem.
EDIT:
I would think so - I installed Ubuntu and I use the account I set up while installing Ubuntu!
No you are not root, you are whatever your username is, root is root. Most distro's do not log you in as root by default, it is discouraged due to how you can mess up your system pretty bad.