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:
If you hold one end of the rope...the rope animates. As a Bonus Action, you can Command the other end to move toward a destination you choose.
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:
If you tell the rope to knot, large knots appear at 1-foot intervals along the rope. While knotted, the rope shortens to a 50-foot length and grants advantage on checks made to climb it.
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:
move toward a destination you choose
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 also tell the rope to fasten itself securely to an object or to unfasten itself...
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.
One Line of This Spell Makes These Ideas Difficult
There is an essential phrase you (originally) left out of your description of the Shape Water cantrip (Elemental Evil Player's Companion, p. 21, bold added).
- You change the water’s color or opacity. The water must be changed in the same way throughout. This change lasts for 1 hour.
All of your ideas seem to involve selectively changing part of the water to one color, and other parts to another color (e.g. changing the water on the legal document to spell out different words in some places, changing the color to be white in some parts of the die and black on others, changing the water to be white on some parts of the card and red on others to represent an ace, etc.). But since "the water must be changed in the same way throughout," this would most likely be impossible.
It might be possible to write a specific message in water on a paper or card, and then turn all that water black and opaque (or clear) as you desired. But you couldn't alter the message to fit your needs. You could argue that you can "cause the water to form into simple shapes and animate" and thus rewrite the messages or cards as you see fit. But "simple shapes" tends to mean polygons (i.e. closed shapes like triangles and squares), and usually would rule out more intricate shapes like writing.
Finally, it's quite debatable whether a wet card would be considered "an area of water" (or whether a wet document would be anything other than suspicious). You've suggested (in a comment) that the card would not be wet, but covered in "a light layer of colored frost." Note, however, that the spell does not give you the ability to melt ice, but only to freeze water. And ice cannot be moved or altered in appearance or shape by this spell: only water.
Best Answer
In 5th Edition a spell, ability or item does what the description says.
The mention of change in style, quality, and that it can duplicate appearance of other (magic, but I assume also non-magic) cloaks makes me believe that if there are cloaks with pockets then this one can take that form.
The length of the cloak is part of style (fashion) change so the cloak could be either a long regal cloak or a short hooded cloak, among other options.