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What year and model car do you have? Any aftermarket things like a radio or remote start?
if the trailer hitch wiring is not the Honda hitch wiring I would start with disconnecting the hitch wiring. I have seen a few issues where the converter box that is part of that wiring goes bad and causes issues. the turn signals are protected by fuse #16 in the under hood fuse box(15 amp fuse)
Did you try disconnecting the trailer hitch wiring yet?
I think you will need to do that because it could be causing issues.
Most likely you will need to remove both tail lights and you will find they attached the trailer hitch wires there. the probably used scotch locks that you can open up and remove the wires.
do you have a battery charger? Sounds like all the checking you have been doing ran the battery down or it has gone bad.
To reset it and get the horn to stop blowing lock and unlock the drivers door with the key a couple times after you hook the battery up and the horn starts to honk.
If it is just the horn that is honking it will not mater
Try disconnecting the battery and them turn the key to the on position. While the key is in the on position hook the battery back up and see if it will start.
When you inset the key do you see green light flashing that looks like a key?
If that light is flashing the vehicle lost the key programming. If you have another key try it and see if the light stops flashing. If the light still flashes constantly you will need to have it towed to the dealer to get the keys reprogrammed.
Thanks for the update. Sounds like the remote start may have been your whole issue.
I may not have been of much help so i am not leaving an option for you to accept my help and pay me.