The horn goes off by itself. The alarm sometimes

Sometimes its because of a faulty charge connection at battery. Check..
If all is ok, and still a problem - look for the fuse to horn/alarm , replace
The FOB and ecu also has issues when parked around alot of other cars, garage doors transmitters/ codes.. in short it could be a variety of options

Sounds like you are starting the vehicle with a duplicate key that doesn't have the chip in it. The theft alarm is tripped when it is started without seeing the chip code.

PJB?

Replace the door ajar switch.

If it is the horn that keeps going off, pull the horn fuse, that should stop it till you figure the problem. The bcm can ground the control side of horn relay as well as the horn switch. I'm not saying the bcm is faulty, I don't know, at the moment.

What your hearing is the factory alarm not the horn. Connect the battery when the alarm sounds go to the drivers door lock with the key and lock and unlock the door until the alarm goes off . Then reinstall the fuse for the horn.,

take the fuse out for the alarm

methinks a stray wire in aftermarket alarm system. Most horns are negative switched (ie they are operated by earthing a wire on the steering column) and would not interfere with positive-switched items such as the hazard lights. Does your car have an aftermarket alarm?

I've had this exact problem before when being jumped off. The fob DOES NOT make it stop. We removed the fuse that goes to the horn. As soon as your battery gets recharged from the alternator, the fob will work again and the alarm will quit sounding. The only pain about removing the fuse is that the cruise control will not work until you put it back.

If there is no original remote try to find a switch that may be hidden but accessible from the drivers seat, if the alarm is no longer needed find an auxilliary wire that may be on the battery or the fuse panel, if original there may be a fuse under the hood, pull the fuse and it should disable the alarn system.