The voltage regulator only handles that you maintain <14.8volts when the bike is running.
If you start the bike with a voltmeter attached somewhere, and then rev the bike up to around 4krpm, it should stay stable <15v. If it goes >15v you need a new regulator. This should have no impact at all when starting the bike, as there is no power being generated yet.
The horn? no, nothing should affect it either way. Most likely the contacts in the handle button is dirty and needs a cleaning. Easy way to be sure is to remove the horn, and plug your voltmeter into those wires. Then press the horn switch a few times and see what voltage you get, it should be a nice >12v and hold solid. If you can test the voltage on the horn with the horn still attached that would be better.
If it doesnt hold solid or doesn't register, you switch needs to be cleaned, or check all the wires are making good connections. If it's all good, then likely the horn is bad.