Can new photoshop be set up to run like CS3?

The elephant in the room here is CS3 is ancient, discontinued & unsupported.  The aging activation servers are gone.  Even if you have an installer and a valid serial number, it's useless because you can never activate the software.  DO NOT UPGRADE YOUR MAC until you're fully committed to all new software, especially on Catalina which only supports 64-bit apps.

In terms of using Creative Cloud, you need to practice to a) get comfortable with the new UI and b) learn how the new features work.  That goes for any software that's 13 years newer than what you've been using. You have some catching up to do. 

Change is hard, I get that.  But you can't go back to how things were.   Set aside some quite time to get accustomed to the new Photoshop.  After 28 days (the time it takes for human brains to adjust to new things),  you'll never look back and probably kick yourself for not doing this sooner. I know I could never go back to CS3.  It's just too primitive.

With the "new" photoshop you can turn off the Window>Application Frame and have documents open as Floating Document Windows (uncheck Open Documents as Tabs), so it works pretty much as cs3 with the exception of the cs3 only Maximized Screen Mode.

If you are running CS3 on your Mac - I doubt you can upgrade your Mac at all at this point. Apple stopped supporting Lion and older OSes. Even if you could upgrade, you won't be able to get to a minimum OS to support Creative Cloud.

Either way, you would be looking at having to buy a new Mac.

As for transferring images from one document to another - the same methods in CS3 still work in current software. Copy, paste, etc. are still there. What method are you using in CS3 that you can't seem to do in Creative Cloud?

»the way one opens and transfers an image or multiple images to create a new image is completely different.«

Not sure what that is supposed to refer to – cloud documents maybe? One does not have to utilize those. 

Please be more specific.