The Curt Custom Fit Vehicle Wiring # 56087 that you referenced will work on your Suzuki Equator even if you do not have the tow package. This product plugs in line with your tail light harness on your vehicle and is not dependent on whether or not you have a tow package.
The easiest way to determine if your vehicle has a tow package would be to contact a Suzuki dealership and give them your VIN number, from that they should be able to tell if you have the tow package.
Typically, tow packages... view full answer...
The recommended ball mount for the hitches for the 2011 Suzuki Equator is a Ball Mount # AMSC2.
The etrailer.com Ball Mount Kit # 989900 that you referenced includes the # AMSC2 as well as a Ball Mount # 2753 that has a 2 inch greater drop. I also includes a 2 inch ball and a 1-7/8 inch ball that way if you had two trailers that have different coupler sizes you could just swap ball mounts and be good to go.
I also attached an FAQ article on ball mounts for you to check out. view full answer...
You have a short between the brake output circuit and the 12 volt accessory circuit. That's what's putting 12 volts on your trailer's brakes at all times when connected.
You'll want to go over the wiring of the trailer looking for any exposed wires touching each other or corrosion and fix as needed. It's also possible that the 7-way is incorrectly wired. I attached a picture that shows how a 7-way should be wired by function. view full answer...
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