If it helps, I fairly regularly fly between Heathrow and LAX on Virgin flights.

For one, they let you take your guitar on as hand luggage if the flight isn't full (and then it gets it's own seat! - no room in the overheads). Last time I went I felt like Jimmy Page with my guitar strapped in to the seat next to me!

If the flight is full, the guitar goes on as 'Fragile'. Gets loaded separately and comes off again by hand rather than down the ramp. You can also ask for additional insurance should you require it.

I've never had a problem just leaving the guitar in it's case as it is. Obviously, the more care you take the better, but try not to worry too much. Sadly, they are more likely to lose it entirely than they are to damage it. Hard cases are for scuffing, so unless you like all your kit brand new and shiny, let it get beat up a bit - it adds character!

I'm assuming it isn't only Virgin that will help you out like this - flirt a bit with the (hopefully attractive) check-in staff.... it all helps.

Mr LordatheStrings - I've always slackened my guitar strings, previous wisdom having been that they'll break with the change in air-pressure. Is this a myth?