Yes, it happens all the time. I designate 3 animals for butchering at a time and wait until the butcher's shop has been cleaned out before designating a new batch.
One reason for hauling not to be performed is the usage of quantum stockpiles: if the feeder stockpiles get filled up, haulers will turn to other hauling jobs before going for the "fill into vehicle" ones, causing things to rot/wither.
Dumping jobs are the lowest of the lowest, and can appear and disappear on the job table dynamically (and there's no indication when all dumping jobs have been performed).
Burrows won't help as DF allocates jobs that are outside the burrows to dorfs, only for them do immediately drop those jobs, get back to find a job, and pick up the same one again, leading to both spam and blocked activity. You'd have to actually reduce the number of competing jobs to steer them to the ones you want by removing hauling categories you want to ignore for now on some or all haulers, and by forbidding competing items of the same category (similarly to suspending building jobs will guide builders to the important one of bricking up the entrance when the building destroying necro horde arrives in the first summer).
The exact details of how hauling jobs are generated are probably not known. The jobs are generated by the stockpiles as they detect items that can be hauled to them, but there are oddities in that logic, as everything isn't listed and it sometimes can take a long time before That Important Thing gets scheduled, despite being the only thing in that category and there are plenty of dorfs who are just idling or performing normal Socialize/Pray activities.