Your symptoms point to a clog. Could be in the toilet, could be in the line. Most smaller homeowner type snakes have a small head, and it may only poke a hole in a bigger clog. Being able to pour water down without it backing up, but flushing something and having it back up points to a partial blockage that gets worse when you add in the waste and toilet paper. A vent is not the problem...a toilet will drain without a vent just fine - it might create problems for other things if the vent does not exist or is blocked, but it won't affect the toilet. The only time a wax ring might be the source of the problem is if there's too much wax and it squeezed into the opening of the drain. Using a wax ring with a plastic funnel COULD be the source of your problem. If it isn't centered in the drain or is too big for the drain, it can get distorted and restrict flow of solids.