Will a cloned hard drive function exactly as the original? If I buy a new hard drive and clone my original onto it (using Carbon Copy Cloner or something similar), will my computer function exactly the same as it did before? Will things like licensed programs not work properly? (I can't think of any reason they wouldn't, but you never know.)
Yes.  I swap out drives all the time... it's basically transparent on a Mac.  I've even booted my MacBook Pro's drive on my wife's MacBook Pro and vice versa.  It takes a minute longer to boot as it reconfigures itself but once it starts you can't even tell the difference.