Thanks for replying! I did finally find what I was looking for-- Suspend is the correct term-- accessible from clicking on the wifi/sound/battery section of the system tray. I'd prefer to have it on the Zorin button menu, but this will do well enough.
Still no luck finding any solution for notifications, though. Judging by thesethreads there might not be one, which could be a dealbreaker for now.
Due to this, as time has passed, my knowledge of the workings of Gnome D.E. (Such as specific details like notifications) has decreased...
I do not know how familiar the different D.E.'s are to you, but Zorin OS Lite does not rely on the need for any extensions as the desktop itself is fully featured. For example, I can select one button from the "start" menu, then I get a menu offering to:
My notification alerts all work properly.
I wonder if Zorin OS Lite might actually be a better fit for you.
I'm not sure about the notifications, as far as I know the notification system only allows for "popups" but won't affect the app icon on the taskbar. Perhaps there's an extension for that but I'd have to look for one. In that regard, maybe ZorinOS Lite may have be a better fit for you although I'll let @Aravisian answer that.
As for the suspend button, this option used to be available in ZorinOS 15.3 when you clicked on "Power Off", as you would get multiple options there besides actually powering off your computer. Not sure why is not there anymore, but with ZorinOS 17 just around the corner it may be something we'll see re-introduced.
Another way you can suspend (which I've personally grown very used to) is to search for it in the Activities Overview. By default you can access it by pressing the Super or Windows key, or by clicking on the button that is right next to the ZorinOS logo on the taskbar. You can modify all of this from Zorin Appearance -> Interface.
I think this depends on the application itself, not on the system. Developers should set a special icon to show new notifications specifically for the taskbar. But maybe the behavior you want is already in its settings. However from inside the application itself you should check if you can turn on notifications, which will then be shown on the notification area. Then, finally, you might want to check in Settings > Notifications if the application has notifications turned on.
Sleep can be set from lock button on Main Menu (the Z logo on taskbar) as Aravisian said or from system tray as you discovered. In both cases, 2 clicks and it's done. You can even set a keyboard shortcut (by default is Super+L).