From the skeleton template (emphasis in original):
Treasure: Generally none, although sometimes a skeleton’s creator arms it with magical armor or weapons.
This is the only thing I can find in the rules which directly addresses the three points raised in this question, and it does not explicitly confirm or deny; so, by a strict rules-as-written reading for all we know arming a skeleton could be interpreted to be a one-time affair. But it's doubtful anyone would rule that to be the case. If someone is going to be that strict in their ruling, then they would also rule that skeletons cannot be made to enter an area first since one must first be somewhere in order to remain there and the only way explicitly mentioned of getting someplace is to follow the controller. Even so, simple tasks such as hauling a log could be accomplished by tying a hauling apparatus to the skeleton(s) and/or zombie(s) — or, if someone wishes to be extremely pedantic "arming" them with the ropes or leads to said apparatus — and commanding them to follow while the controller walks the route.
Outside of a strict rules-as-written interpretation, this GM allows basic environment manipulation and basic movement commands — including basic conditional logic such as that required to attack certain kinds of creatures, and relative position — if for no other reason than to avoid the outcome of every animated corpse of a group of several attempting to follow the caster at the same distance and constantly bumping into each other. Such basic commands would include the tasks you listed, and others that expand on things we know they can do: they can exhibit simple conditional logic, they can grasp and wear items, they can move relative to a designated creature or position, etc.