The rope of climbing is described as:
This 60-foot length of silk rope weighs 3 pounds and can hold up to 3,000 pounds. If you hold one end of the rope and use an action to speak the command word, the rope animates. As a bonus action, you can command the other end to move toward a destination you choose. That end moves 10 feet on your turn when you first command it and 10 feet on each of your turns until reaching its destination, up to its maximum length away, or until you tell it to stop. You can also tell the rope to fasten itself securely to an object or to unfasten itself, to knot or unknot itself, or to coil itself for carrying.
What defines the motion of the rope? Can it ascend vertically into the air? Can it horizontally arc across a river without touching the water?
How much lifting capacity does it have? The text says the rope holds 3000 pounds. How much weight can the rope hold as it climbs into the air? For example, if the rope climbed 50 feet into the air straight, can it only support itself? Could a piece of paper or an apple be attached or would the rope collapse? Or conversely, could someone climb the rope (like Rope Trick)? Could a 3000 pound attached basket of people be moved 10 feet a round across a river?
Could the rope be used to grab someone unconscious down in a pit, lift them up to the edge of the pit, then have the rope continue to climb up to a ledge above while still attached to the person thus lifting them?
Best Answer
No, it can't lift a party across a river
It can create a single rope "bridge" that the party would then need to climb/walk on to cross the river.
Holding vs Lifting
The Rope of Climbing (DMG, 197) does not Lift anything per the description. What it does is:
This is clear in that you are holding one end of the rope while the other moves towards the destination of your choice. Once the end arrives at it's final destination, you (and others) would then still have to climb the rope. It does allow for easier climbing to limit risk by:
It is not meant to carry you up. It simply removes the requirements of using a grappling hook and still requires someone to climb it. If it was meant to carry you up, there would be no need for the climbing mechanics or the knotting to assist in climbing.
As for horizontal/arc movement, that's perfectly acceptable as well. There are no stated limitations on how the rope moves 10' and in fact does state:
This allows the rope to move in any direction you choose.
No language on what happens if you overload
Frustratingly, there is no language on the results of putting more weight on than the rope can handle. Other magic items do provide such effects, but it is not described in the DMG and is therefore under the purview of the DM. But given that it doesn't say the Rope is destroyed, it's more likely that it just can't continue to lift/move if overloaded.
Get creative
There is language in the description that just begs for creative thinking (emphasis mine):
You can not only use it like a grappling hook that'll always find it's mark, but it can be a 60' long extension of your hands as well. Need those keys? Rope gets 'em. Gotta pull that lever across the chasm? Send the rope! It is not just a means of getting from Point A to Point B, but also a means to bring/interact with things at Point B.