While I don't pretend to be an expert on all of this, here's a few thoughts.
1) If the valve was stuck open, I have to wonder if that wouldn't eventually cause the train to stop from low air pressure.
2) AFAIK, a continuously blowing horn constitutes a non-working horn, just like one that won't blow at all. An engine with a non-functional horn cannot be operated period according to FRA regs.
1. No-The horn uses very little air comparitive to the output of the compressor, so it could keep going until the loco runs out of fuel, and then probably take more than an hour, to use up all the air. And, the horn will not affect the brakes-there are two separate air resevoirs on locomotives, with a one way check valve between them-i.e. the air brakes use air from one, auxilliary equipment like horns, use the other, and even if it's pressure drops, it won't reduce pressure in the other, affecting braking or other critical equipment. To silence a stuck horn, there is a valve that can be shut to cut off the air to the horn- on older locomotives it's in the control stand-Just turn it, and the horn is cut out and sileneced. Problem is, it usually requires tools to remove the panel to access it-Some engineers carry tools, or a leatherman/Gerber like tool, and can get in, others don't. No tools, no access. I'm not sure where the cutout is on newer desktop control stand locomotives, but I assume it's also similarly inaccessible. Mechanics don't want engineers messing with stuff....
2. Yes, and no-Horns must be operable on the lead locomotive when departing a terminal. En-route, if the horn fails, as in, doesn't work or is stuck on, the train can continue, by ringing the bell continuously, and the train must stop at each crossing to have the Conductor/Trainman flag and protect the crossing, unless the crossing is equipped with gates that are fully lowered, or there is no traffic stopped at, or approaching the crossing, in which case the train can continue moving, and then get repaired or switched around at the next terminal.
There are a few ways to correct this, either cutting out the horn if you can get to the cutoff, or if the horn is stuck on, and can't be shut off, and they're going to send someone to fix it right there, the engineer could secure the train and close the Main Reservoir angle cock between the lead, and any trailing locomotives to keep from charging the lead air system from trailing unit compressors, shut down the locomotive, or disengage the air compressor, and bleed off the air from the Main Resevoir manually. Takes just a few minutes, and will shut it up. In that case however, the train can't be operated, because the brake system on the lead loco is now no longer functioning properly, so they'll have to come to fix it on the spot. Which, is much better than having to run a train with the horn blowing non-stop!