I'm looking to reinstall Windows on my SSD but I really don't fancy reinstalling a ton of software. Is it possible to backup the entire registry of an old Windows installation to a .reg file and then restore it on the new one on the SSD and just copy over the Program Files, ProgramData, and AppData folders to avoid reinstalling everything?
Windows simply wasn't designed with this sort of usage scenario in mind. I've been fighting such cases since about 1994, when I first started using Windows NT 3.51, and I don't think I've ever gotten any kind of naively-cloned Windows install to start up cleanly. It's why there is so much specialized Windows drive cloning software on the market. We we wouldn't need all of that if a simple disk duplication worked and the OS came with the tools you needed to do that.
The biggest thing you're going to be missing out on with your plan are permissions. Just like files on disk, every key in the registry has an ACL, but
.reg files don't include security info. If you don't copy all the ACLs over, the best possible case is that you lose a bunch of security because you make all keys readable and writable by normal users. More likely, you'll end up with registry keys that the OS refuses to accept; there's an excellent chance that the OS will just blue-screen on first boot.
Then you have other details to deal with, like unique per-system and per-drive SIDs.
Bottom line, either use special cloning software or be prepared to do a lot of manual re-work.
Actually you'll have more of a problem copying over the files than with copying the registry. The registry is just a few files on the disk that can be copied normally (e.g.
%SystemRoot%\System32\Config\SYSTEM). The problem is that when you copy files in general, you can mess up features such as hardlinks, junctions, extended attributes, permissions, etc... the only way to get that right is to get an image of the drive and restore it.
For decades, I have been installing all programs on a second hard drive. Over time, I have found that most programs run without problem even after a new installation if I simply launch the appropriate *.exe file of the respective program from that program's installation folder.
As for the settings, you could try to simply copy the entire user folder of the old installation back to the user folder of the new installation using a live CD. After this, also most of the program settings from the previous installation should be present again.
Also, there is an installation tracker called Martau Total Uninstall, which I have also been using for decades.
Total Uninstall can read out all files and registry entries of an installed program even after having installed and used the program, and export them as a ZIP archive. This archive can then be restored on a new Windows installation, or even on another PC, in such a way that the program and all its settings work again.