GP won't start!!

Hello,
My 00' GP se v6 3.1L 129km+ is having a difficult time starting unless I press on the gas pedal. I recently took the car to Autozone to have it check out and consequently on the device that reads or troubleshoot parts shown that I have a dead battery. Yesterday I purchased the correct size battery for the car (group 78) and once the new one was in I tried firing up the engine but I experience the same result, a sluggish or deading engine trying to start. Today, I took the car back to Autozone to have my battery rescan and scan other parts that may be causing this problem. The results shown that everything was good: battery, alternator, and starter. I was baffle because now I have no idea what may be causing the car to have this rough, grinding, or sluggish sound when I try starting the engine unless applying the gas pedal simultaneously. The staff at Autozone said it may be a faulty ignition switch. I have no idea where to start but I am guessing it may be a corroded or damage spark plug cable or coil because every time I shut off the car after driving it I always smell and hear as if something is burning.

I think I found the culprit for my troublesome starting issues. In the picture below you will find a severe rusty, corroded #4 ignition coil and #6 wasn't too bad. I decided to scrap off the rust buildup off both #4 & #6 especially and then try firing up the engine and it started up a lot better than before. I am not sure if a faulty ignition switch would cause the engine to have trouble starting but I believe this is one of the issues. I almost forgot to mention whenever I try starting the car I would hear a clicking noise as well. Could it be a starter trying to ignite? I'm clueless.