Can a bad positive battery cable cause transmission problems?

Twice when putting my car into reverse the vehicle shifted harshly into gear (only when car was cold and had just been started). And once while breaking at a stop sign my Ford Fusion jolted for a split second (it felt like the transmission dropped out of gear violently). All power remained on, but the odometer displayed dashes for a few seconds and then returned to display the accurate mileage reading. I promptly took the car to my local Transmission shop. The technician went to test drive my vehicle and for the first ever it would not crank over. Another technician came out moved the battery cable around and the vehicle started. I went ahead and had the transmission shop service the transmission and check for error codes. No error codes were found.

After the transmission service I have no longer experienced hard shifts into reverse. The vehicle continued to run perfectly for a couple hundred miles. While using speed control and driving approximately 60 mph (for a steady 10 minutes), the car violently shook again for a split second and the speed control turned off, the speedometer also dropped to 0 mph and then quickly bounced back up to the correct speed. I continued driving the car for another few weeks and the transmission would exhibit strange behavior. Twice when accelerating from a dead stop it felt like the transmission was slipping (only under 7 to 10 mph). All these (what I assume to be transmission problems) happen for a split second and occured a total of 8-10 times over a thousand mile range.

Then the vehicle experienced starting problems. All lights would come on, but the engine would not crank. I had my wife try starting the car while I fiddled with the battery cable and the car started right up. The battery cables were tight and there were no visible signs of corrosion. However, the cables do not appear to be stock, the positive cable is split two inches off the terminal and held together by a bolt and nut. I unbolted and cleaned the connection that I beleive to be non stock. A very small amount of corrosion was removed.

The car then ran perfectly once again for about 350 miles. I experienced no transmission problems or starting problems. But gradually the transmission acted up twice like previously stated and I'm now having the same starting issues.

I'm on a tight budget and I'm debating if I should take a chance and buy new battery cables for 90 bucks and hope that fixes my starting and transmission woes. Or take it to the dealership and have them investigate the problem for about the same cost. I've already spent $160 on a tranmission service and diagnosis at a transmission shop, which resulted in zero error codes and them telling me nothing is wrong with the transmission. Can a bad battery cable cause transmission problems?