I bought Photoshop CS3 back in June of 2008--if I upgrade my laptop to SSD and would need to download CS3 again, would I be charged a monthly fee under the new "Cloud" system?
No. CS3 has nothing to do with any Cloud, it's an old, obsolete, unsupported application that Adobe no longer sells. It has never been part of he cloud.
Your CS3 is a perpetual license that allows you to install the application on two computers that will not be used at the same time. If you should need to install it on a third computer, just deactivate one of the other two.