Is that supposed to be the drain test you are showing? Hope not, because it's wrong and you are getting close to blowing the meter. I have no idea what you are showing in the meter picture. Please explain.
After that I will run you thru the test now that I see your meter. I would like to see a little better picture of meter by itself.
As far as the horn going off, that ain't good. So show me the relay wiring and explain exactly where each wire goes.
"Before that, I had the positive cable hooked to the battery, and the negative cable disconnected. I put my mm on 10A and the red lead is in the 10ADC hole. When I touch the black lead to the negative battery post, I was hearing this clicking sound."
Correct me if I'm wrong but this sounds like you had the meter red plug into 10A jack and dial on 10A .............and red lead hooked to positive battery post and black lead to negative battery post? Yes?
Where was the clicking sound coming from? Horn trumpet, horn relay or meter?............... or a cricket?