I took it to a mechanic yesterday. Today he told me he couldn't find the problem yet. We'll see.

Thanks a lot for the help.

PS. I decided to reset the system by unplugging the battery yesterday morning. Didn't check the manual and removed the positive end first, and the alarm was triggered (without the horn). When I plugged the battery back (but without the key in ignition), the horn joined the weak alarm sound, couldn't get the keys to work, unplugged the battery again, waited for the car to be towed for an hour while listening to that annoying alarm bell. Then plugged the battery back and this time the key worked and silenced the alarm. Just remember to check your manual for unplugging the battery