You won't see a strong discharge on the ammeter while cranking because the starter is wired directly to the battery and that current doesn't go through the ammeter. It's possible that the alternator is supplying current at crank speed, which would show as a positive.

The horn button should ground the wire to the relay, and if it doesn't, the relay won't work. You could ground that wire at the relay to see if that blows your horn, but you have to determine why your horn button isn't supplying ground when depressed.