Sorry for the long title. My question is basically if my Ubuntu software bought from the Software Center will be left to download if I'm logged in on another Ubuntu (other harddrive)? Or do I need to make a copy of my other Ubuntu and copy the software that way.
If I'm understanding your question right, you are asking if you can download multiple copies of software that you bought using Software Center. You can. Just go to File > Reinstall Previous Purchases... while in Ubuntu Software Center.