I'm aware of applications such as this which allow windows to be "halved", allowing the screen to be divided up. But are there any apps out there that split the Mac screen within applications and within windows? I'm aware that split screen editing is available in Microsoft Word and Excel, but is there an app which is allows a universal split screen across all applications?
Put the same application on both sides, then select the window that you want in the left side and the other window in the right side.
Most Mac apps allow you to create duplicate windows of the same document, if they have a "New Window" menu item in the Window menu. You can then arrange these two windows of the same document (views of the same window, if you like) as you wish.
I think that's the best you're going to get natively without some additional software.
Long press the fullscreen button, and then you will figure out the rest.