As the title suggest, can I just copy and paste an application to my new MAC instead of downloading them all over again? Or is there another way to do this?
Try the migration assistant first. It will move everything you need from your old mac. (OS 10.6.8 or later)
Many applications have data in the /Library/Application Support folder or the ~/Library/Application Support folder as well as prefeferences and other scattered support files that would all need to be moved.