I thought airlines just preferred to have money for the seats. Whether occupied by a Strad, cello, dog, or specimens doesn't matter as long as they have $$ for the seat.
I'd have thought so to, but I guess not.
There are a bunch of regulations about cargo occupying seats in the passenger compartment, which have to do with safety or the appearance of safety. For obvious reasons, cargo can't ride in a seat where someone would have to climb around it to get out, but the regulations go on to be more restrictive than that. For example, see
http://www.cello.org/Newsletter/Articles/flying.html. There's a bunch of other rules for pets in the passenger compartment, by the way.
I don't know whether baggage or customs people will read notes that are taped to the box, but it probably doesn't hurt. I used to travel with a piece of hand-carried mystery equipment and usually tried to carry a letter on university stationery saying what it was and that it was university property, the latter also intended to deter Customs from charging duty on it.