I am a current Gallery Project user and thinking about moving on since the project is going into hibernation. But I need to know if Zenphoto can do what I have Gallery doing currently.

I have over 600 users and 2013 hi res images on my site. All users have to login to view and download my product images in multiple sizes from thumbnails to original file sizes.

I also have the image names showing under each thumbnail in every album.

Can this be done with Zenphoto?

I am not sure if any tried 600+ users yet but possible it is. For the downloads you probably need to do some theme work.

General article about user rights:
http://www.zenphoto.org/news/an-overview-of-zenphoto-users

I also have the image names showing under each thumbnail in every album

No official theme does it so you would have to change the theme chosen a bit. But if you have a bit web knowledge that is not that hard actually.

Zenphoto can do a lot and is quite flexible but not all per click. I would say best try a bit around on a test install. Also read a bit on the user guide. We happily try to answer further questions.