Car starts by itself even when the keys/fob were 100 miles away

Try locking it with the fob, then unlock it with fob. If the antitheft is engaged, the light will stay on or flash.

key fobs have to be programmed to the immobiliser in the car. Take it and the car to a service dealer and have it programmed properly.

If this only happens when the key fob is close enough for the car to "hear" the signals transmitted from it then chances are the fob is failing. The FOB will have to be taken apart, cleaned and inspected for contamination/corrosion/stress cracks.
If the car does this on it's own even when the FOB has no battery or is too far away to be heard, then the problem could be internal on the TWICE (Theft Warning Module) or a power lock switch.
Since you said the horn beeps I would say it is most likely a FOB inadvertently transmitting signals even when you're not pushing buttons. Repair or replace the fob and you should be okay.

Same issue here, Sonata 2011 GLS and no key found (but more and more often at winter time) At dealership they said....its 100% computer problem, we can change it for $900 but we dont guarantee it will work normal again. Replaced 2032 battery 3 times made no difference so i thought its a power issue somewhere. i left my car on parking for 2 days for test. Opened door and saw it again...key not found. I connected a booster cable from running car to battery and left it for 10 min to charge. AFter 10 min i had a surprise...key found and car started easily. So its the Hyundai Factory car power design issue. Now i need to change my alternator to cover my theory. thx. ___

try the other fob.
the buttons on fobs can short.(its just a plastic membrane with carbon pad in the fob and can go bad.(cracks or crumbles inside...........)
so put the suspected fob far away from car, and use the good new fob. work now, and a new FOB battery, too.
if you only have one fob, now's the time to get the spare.
no fob lasts forever. not not ever.

lost both FOBS?
be happy, mine i must keep 3 working.... or cant make my own.
rule 1, never have 1 fob. or get stranded, , you will.....
so does other fob work?

1: check all fuses. both boxes
2: use only the key fob, no other keys on ring, for the test.
3: use the 2nd fob same way , now. with other key far away.
4: make sure battery voltage is 12.6vdc, 10.7v the IMMO goes belly up. (any $10 voltmeter , works)

5: have only 1 fob (transponder-key) near the ignition.
6: do not have other keys near Ignition switch. even just plain keys.
watch out, some after market keys are transponsers but look plain.
I have one for my car. button less fob. very trim, but is transponder key.
7: for sure not other transonder keys from other "cars or anything"

The Lamp means, your transponder key is dead. or scrambled.
buy a new battery for fob. the operator guide tells you what to do.
lost that?
on mine i put in a fresh coin cell and bam, it works.

on newer jeeps, if the IMMO thinks your are hacking the system.
it bricks the IMMO for 24hours, sucks huh?
for example, using a key from another car , in some odd way.
or doing same and battery is weak or remove from under hood.
it thinks its being attacked,, and goes stupid for 24hrs.
i dont think yours has this, , but mine does.

lock all doors and trunk then use key fob to lock all doors and set alarm ,then wait few minutes and unlock vehicle and try to start

I just changed the battery in my fob for 2011 Journey Crew, with the battery removed ( from fob ) I did start my car by pushing the start button whith my fob... so the theory is tested and true.... also the battery you'll need is CR2032 available everywhere... and removing the battery will NOT remove the programming of your fob ( also tested )... so no need to visit the "steelership" Hope this helps....

If the key and the fob have separated, you should still be able to start the car by holding the fob near the ignition when you start the vehicle. This will allow your car to read the immobilizer code from the chip in your fob. You can do this temporarily until you can get a new key. Once the car starts, you can move the fob away - it only needs to be there while starting the car.