Landing gear pins were mentioned elsewhere, simply because if I have a problem involving them, then it's my fault. That's far from the only issue one can have with the landing gear...but it's the only one that's a bit hard to explain.
There are lots of safety systems tied up with the landing gear. Various switches have to be made for the aircraft to decide that it is in flight, and for it to even allow gear retraction. These switches can also involve lots of other systems (for instance, reverse thrust, ground spoilers, even engine idle speeds). An aircraft that thinks it's on the ground, when it's in the air, or vice versa, can be quite a handful.
During the retraction sequence, various parts of the gear tilt, doors open/close, and all of these have to happen in the right order. If, because a sensor has failed, the system detects a fault, then part of the gear retraction sequence may fail, or the whole sequence may stop. For instance, partial retraction would be the most likely outcome if a wing landing gear tilt sensor were to fail (if I remember correctly, there are two per truck, and they must AGREE). That's much better than the alternative, where attempting to retract gear that were (for example) at the incorrect tilt, would most likely cause damage to doors and the gear cavity, and perhaps even jam it in place.
As for a PA...well you get to it when you can. This is most likely a two pilot operation. They have a lot of things to do, and, quite simply, a PA is not high on that list. With an extra pilot, you'll most likely use him to make PAs, but it's a distraction with a small crew.
Would it not depend on how heavy the aircraft came in, as I'm guessing that dumping fuel for an hour only, puts it overweight for a landing? There may have been all sorts of checks that needed to be done if so.
An hour would be plenty to dump to well below MLW in a 747.
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