Q: Gurgling noise while brake and pumping feeling in reverse

The intermittent illumination of the Check Engine Light (CEL) should be resolved by getting a CEL Diagnostic and then repairing the fault that is identified. Such a diagnostic might provides clues as to the other faults you are writing in about, namely the jerking in reverse and the unusual noise. Causes behind transmission shifting issues include low fluid level, valve body faults, internal hydraulic failures causing fluctuating pressure as bands or clutch packs are actuated and so forth. All this can be evaluated in a shift diagnostic. The origin of the noise would have to be evaluated first hand by a certified Mechanic because there are a variety of hydraulically operated systems on a car and so a Mechanic would have to hear the noise first hand to pin this down for you. During a CEL diagnostic though, obviously the Mechanic would be right there to hear and identify the origin of that noise.