I had a similar issue on my old equinox. 2007 and above the door chimes and turn signal sounds go through the drivers side front speaker. On 05-06 it goes through a speaker on the instrument cluster. Check the wiring loom in the door harness, chances are is that it is chaffed. Another possibility is the head unit acting buggy. My 08 equinox LS has the basic, BASIC radio that only displayed time and nothing else. If you have the LS model chances are you have the same one. I noticed the same issues, and my door harness was fine. Changed out the head unit for a cheap score from the junkyard with a Pioneer double din ($15 with the adapter and harness, literally a 20th of what it would cost to build that from scratch) and the issue was completely gone. My tweeter was the only thing that had worked for the longest time. Also, maybe check your speakers themselves for any damage be it to the speaker itself or the connections.