Can a old battery cause an issue in automatic transmission

Not likely that a new battery will fix these issues. The alternator may be the cause of the battery light. Without knowing more I would say your engine may need a tuneup.

I just replaced the plugs, wires, and icm in april. The fuel pump and fuel filter was replaced in may. I just changed the oil and transmisson fluid 2 weeks ago. And its been hiccuping and shifting kinda hard before all that.

Is the check engine light on?

Yes says its an evap leak? Bought the new gas cap cleared the code it came back a week later. The guy at the dealership (that sold me the gas cap) said the evap leak was because the little metal flap in the gas tank hole wasnt closed all the way.

Is there a way to test my alternator? Like there is no signs of a bad battery or a bad alternator. Thought i had a transmisson issue until i realized that like every time i stop and start again as tye car is going into second gear the volt warning light flashes the car kinda hicuups slams into 2nd gear then goes away.