Key-off battery draw: when you shut the car off, and lock it, there are a few electrical consumers that stay energized. Some for only a minute or two, and some longer. Typically, only the alarm system stays on for an extended period of time. If anything else does, this will draw down on your battery. If you have a multimeter with a mA reading, then it's something you can do yourself with the following procedure:
Open your bonnet (engine hood), and then manually re-latch the latch using a screwdriver or similar (we're re-latching so that the alarm computer will think all latches are closed). Open driver's door, and also re-latch. Disconnect negative lead from the battery. Set your multimeter to the A or mA setting, and place your meter leads between the battery cable and battery post. You should then see how much current is drawing on the battery with everything off. Write down, or take a note of the amount of current (ex: 550mA). Then open your fuse panel cover on the left (driver's) end of the dashboard. Start with fuse 1, remove the fuse and look at the meter. If there is no change, reinstall, and move onto fuse 2. Repeat this process until you see a significant amp drop on the meter. You will then know the circuit that is drawing down the battery. Look at the fuse legend, find out what is on that circuit, reinstall the fuse, and then unplug any electrical consumers on that circuit until you find out what specific component is drawing on the battery by unplugging them one by one until the amp reading on your meter drops below 50mA
Regarding the horn question, this circuit is only live when the key is in the on position.