Whether the paddlewheel rolls up (Fig1) or down (Fig3) the cable or neither (Fig2) depends if the force of the water is above or bellow the point where the cable leaves the spool. In the original picture I drew the paddlewheel would therefore roll up the cable and not down as indicated. So will this design ever be able to travel DDSFTTS? (S = Stream)
My intuition tells me that it's harder to get this to roll upstream than you think, although I can't construct a clear argument based on the forces as suggested by your drawing. It could be that this is because they are correct in perfect conditions, and my intuition is drawing on the peculiarities of real water-air-floating-object interactions. In Fig 1, for instance, if you imagine taking away the paddles, you could argue that the water is still impacting on and dragging the base of the cylinder, and that is below the cable, hence it should still wind itself up the cable, but it screams at me that it'll do no such thing and you'll be running off downstream in no time to retrieve it. That may be wrong, or it may be that in real conditions the natural changes in waves or the air currents over the water would tend to unwind it. It may seem unfair to remove the paddles and expect it to still go up-wire, but it's to make the point that there are some weird forces here. If that was fast water pulling on it, with or without it turning, a bow wave is going to ride up the upstream side, too, tending to push it downstream.
Indeed, if you put something like that in what appears to be a fairly still lake, any slight motion will unwind the cable, simply because there is a force of tension that can cause it to unwind, but you can't push a string, so it won't rewind itself very easily. In a turbulent stream, there might be some of that going on.
It's sometimes useful to think of extreme cases to see if it becomes more obvious, so what about this:
That feels like it's got the best chance of going upstream. If we held it at the centre on an axle, of course it's just a waterwheel, and the cable does resist the force applied in a very similar fashion to a waterwheel.
I don't know if it can go DDSFTTS, or if this is the best sort of 'gear ratio':
If you imagine the cable winders being just fairly thin wheels, joined by a long thin cylinder with fins on it, then it seems to me that the whole thing is going to be forced downstream at some fairly large fraction of water speed. My thinking is that as it does so, the vanes are forced to paddle and, although they seem to be paddling slowly by that gearing, they keep providing a little extra thrust, but maybe I've reduced them too much. I don't know. It's a little bit like the vanes on a wheel - at half the radius, a vane below the axle slows to 1/2 the speed of the wheel, so if a windcart propelled by that method has reached windspeed in a tailwind, and can't itself be pushed anymore, the vane can still gain thrust. The point at the very edge slows down to zero as it touches the ground, so a vane there could get pushed whatever speed the cart is going. Against that, there is the lever distance (being zero there), which is relatively easy to think of in the case of a mechanical wheel, but in this case it's harder. Can we think of the winding cylinder just like the wheel circumference, like the thing is rolling downstream laying cable as it goes? That seems convincing, and then my paddles here are too close to the 'axle' to be of much use. They have a great lever distance, but are moving almost at waterspeed, so the water isn't moving past them much. Yes, it looks like my first guess was a bit off. Your Fig 3 looks more likely, or perhaps somewhere between that and mine.
Interesting question. It has some strange similarities and differences to that wheel-with-vanes problem. On that, we need to get the vanes out of the way as they go forwards, so that they're not slowing the cart. Here, we aren't bothering about it so much because they're moving in air instead of water, and the water applies more pressure. However, if this thing is going at some fair old whack downstream, that might still be a problem.
What do you need it for, to send downstream faster than the stream to tell people downstream there's a crazy engineer upstream and they need to watch out for strange vehicles and taut cable coming downstream? Oh no, that would be DDSFTTV. You could warn them of floods coming, or dead goldfish.
Oh no, now I'm confused! I just noticed that I'd put "sin-1" and then "arc-cosine". I'm pretty sure it was supposed to be the arc cosine in mine. Yep, adjacent & hypot. I checked yours, Dan, and you've got arcsin! Copying my wrong answers? 
