Is Creative Cloud manditory>

Some have reported on other threads that they have uninstalled Creative Cloud after installing Photoshop, but I can't vouch for that personally. I use it to get to fonts, for file sharing, and to update my applications.

~ Jane

You download Photoshop from Adobe’s servers and use the application(s) in the usual way on your computer. There’s no difference in use than if you’d installed it from a disk. Adobe checks, transparently via the internet, every few weeks that you’re up-to-date with your subscription. An advantage is that regular updates can easily be offered.

Not sure what notifications you mean? I don't get any.

The CC desktop app is required to install/run/update the applications, but it runs in the background and you normally don't see it.