If its burping that means that the vent is non functional either clogged or the vent pipe is below the fluid level in the tank... Since you ran a hose down the vent and the problem does not occur on day one then its a real possibility that your notion that the pipe was inserted to low in the tank... Or has migrated downward.
Can you see the black tank to know how tall the tank is? If so one could use a tape measure to measure from the top of the vent pipe to the bottom of the tank, then pull the tape up slowly trying to catch the bottom of the vent pipe...
Is the plumbing 3 inch or is it smaller? Can you tell how close to the dump valve the vent pipe enters the tank... The reason for asking that if outlet and piping is 3 inch then an arm encased in a vet sleeve might to remedy the problem simply
The leaking on day one is another issue..Guess on that one I would check the torque on the bolts securing the toilet to the flange. also possible the seal was not correctly installed... Or the seal swells once it get wet.
JohnnyT