Just that. Can an Unseen Servant be moved around if ordered into a pocket, backpack or bag of holding or if merely ordered to "hang on" to the caster?
It's not a creature. It's a spell effect that emanates from the caster. So no, it can't be carried.
So is it like a Mage Hand effect, which always remains with the caster except in the event where it is carrying something and cannot move its load as quickly as the caster is moving, in which case it ceases to exist? If so, then carrying it around is altogether unnecessary; it's just always there.
It's not a creature. It's a spell effect that emanates from the caster. So no, it can't be carried.
There are more rules saying you can carry non-creatures than creatures. Carrying a creature would imply they may need to make ride checks.
Actually, you could command the unseen servant to ride an animal, as long as it doesn't need to make a ride check above DC 10.
A wizard and his unseen servant mount a horse. The horse gallops off. Would the servant wink out if the horse exceeds the servant's movement rate ?
A wizard and his unseen servant get into a canoe. The canoe is carried swiftly downstream by the current. Would the servant wink out if the current exceeds the servant's movement rate ?
I see the speed limit as the fastest the servant can move itself, not how fast the servant can be moved. So long as it remains within the prescribed proximity to the caster, it should be fine.
A bag of holding, as an extra-dimensional space, would probably not qualify as remaining withing range.
If you attempt to send it beyond the spell's range (measured from your current position), the servant ceases to exist.
This is a really oddly written sentence. Technically, you could tell a servant to stay in an area and then just walk away from it. You just can't intend to "send" the servant away from you.
You could even outrun the servant and tell it to catch up, as this would not be an "attempt to send it beyond the spell's range".