usually they can check your luggage all the way through - but double check at LAX when you check your bags in.

I would go straight to the gate, even though an hour sounds like a while - you're better safe than sorry and some airports can take a while to get from one end to another. Don't forget that those are departure times, not when boarding starts - so its probably more like 30 mins from scheduled landing to when boarding starts.

I don't know the procedures at those airports, but I once had the last flight out to HKG from Singapore and while waiting in line in security at the gate an official cut in front of me and announced that it was all closed since it was now 20 mins till departure, gate closed. Watched the plane finish boarding, and taxi away. Not happy.