I have no idea what the hole is or what function it serves, but today it began to emit water. Any appliance in the house that uses water makes water fill the pipe and eventually spill over. A little earlier I saw bits of toilet paper come through. There hasn't been a heavy rain as of late. I have no clue what the pipe is; I have attached an image. Any help would be appreciated greatly.

That opening looks like a clean out access without a cover. The expanding clamps on the wasteline should also be exposed better so we can see what it is and what it does.

Best Wishes

That looks like what remains when someone knocks the center out of a brass cleanout plug and attempts to clean a drain through an improper access with undersized equipment. Get a professional drain cleaning company out there to clean your main sewer line. I recommend a professional company because they can get the old plug out, clean the line out, and install a new plug to stop any sewer smell. In addition, they should guarantee their work.