My toilet bowl fills with water but will not allow the water to flush out. However, when i allow the toilet to sit untouched for a period of time, the water will drain out on its own. Then, when i flush it, it fills up with water but will not flush still. I have plunged, taken the tank apart AND snaked. Does anyone have any suggestions that may save me from hiring a plumber?

THe tank has an "odd" type of hardware that i've never seen. its not a ball and flap but more like a cone type contraption that lifts and drops very quickly.