I must have done hundreds of Santa Fes by now (I do dealer work), and my co-workers and I have gotten them to go pretty quick and easy.....it is a simple car. I will try to answer all of your questions.
--Your post says you're in the U.S., right? And you didn't mention what year your Santa Fe is, but I'm guessing it's not the all-new 2007, is it?
--The U.S. model of the Santa Fe, 2006 and older, does not have an anti-theft transponder key; no bypass is necessary. The 2007 is mostl likely to have a transponder key, and factory remotes that DON'T work when the engine is running.
--That's cool that you're doing it for pet cooling. I have a 1-year old Rat Terrier named Bonnie, who goes with me everywhere, including to work.
--As far as what buttons you're going to press on your remote, and in what order, that all depends on the brand.
My Audiovox stuff is Lock, Unlock, Lock.
DEI gives you one wire, and you hook it to whatever you want, and you can go into the programming menu and set it so that it requires 1, 2, or 3 pulses, your choice.
A typical scenario with the DEI piece would be to connect it to the wire that triggers the power-lock actuator inside the driver's door (you can catch this wire in the driver's kick panel someplace)..........and program the unit for 2 or 3 pulses, your choice..........and then you'll just press Lock on your factory remote either 2 or 3 times, and the car will start.
(Obviously you wouldn't want to program it for just one pulse, because then it would start every time you lock the car!)
--As far as what wires you would connect.....I'll try to give you a quick, non-detailed rundown, for the 2006-and-older.
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--Six wires in the thick harness at ignition switch. Two constant, one ignition, one starter, two accessories.
One of the accessory wires comes on when the key is in the first "Accessory" position; this powers the radio.......I leave this wire unpowered, because I don't like the radio on during remote start.....but it's up to you.
--You might see various diagrams telling you where the keysense wire is........first, it is not a necessary connection..........second, if you do connect the keysense, the factory remotes won't work when the car is running.......so don't.
--There's a white plug a little to the right of the fusebox; + trigger parking lights is GREEN/ orange.
--Brake wire is GREEN/ black in the plug at the brake pedal; easy and plain to see.
On some of the Santa Fes, though, the brake wire can also be found immediately next to the parking light wire.....I like to get it there if I can, because it looks neater.
--The wire that triggers the driver's doorlock actuator, well, I don't know where that is, because I haven't put an add-on type of remote starter into a Santa Fe before.........look for a wire diagram for your car and find the "lock motor" wire; that's what you want........it'll turn into a positive when you hit the power lock switch.
If all you want is a BASIC remote start install, with no concern for safety features, that's it; you're done. It really is an easy car. Program for voltage-sensing mode, 0.8 seconds crank time, and it'll work fine.
Did you also have a question about the Hyundai's factory alarm? Nothing to worry about; starting the engine will not set off the alarm anyway.
If you don't mind a little more work:
--The factory hoodpin wire is available inside the car.....I think it's pink, on the left side of the fusebox.....you have to unbolt the fusebox and shove it aside a little to gain access, and even then it's a little awkward.
In any event, it's far easier to take advantage of the factory hoodpin wire, than to fish your own wire through the firewall and install that crappy aftermarket pinswitch that'll rust out anyway.
--If you want to connect to the car's tachometer circuit, which is more reliable than voltage-sensing mode, that wire is available inside the car too.
Remove the long panel on the side of the center console, to the right of the driver's foot. Maybe four or five screws.
You can then plainly see the engine computer there. I THINK tach is WHITE/ black.....but it is always in a different location, depending on which engine and transmission you have........so I always have a hard time finding it.
The DEI wiring sheet lists the exact pin # and location of the wire, which might be of help to you.