Sounds like you got a good idea what you want.
I think it should work fine as you discribe it, sort of a paddle wheel boat but the water turning the wheel instead of a motor.
I think most the questions you asked though will depend on the water source. The deeper you go the more resistance your gonna get with the paddles entering and leaving the water, but the deeper you go most likely the stronger the force of moving water which is what you need. As for the paddles, I would think maybe a curved shape would be better than flat. It might grab and hold the water better entering near the top of the river where it would still be at a steeper angle, near the bottom of the stoke where it is vertical it would be working well, and as it raises near the top again being curved may have less drag and resistance as it leaves the water.
I have noticed sometimes there are 4 types of rivers. Those with top moving water, but the lower water is fairly still. Under tow rivers, top is fairly still, but alot of currant deeper down. Rivers that move fast at all levels, And rivers that don't move much at all!
I often go canoeing on a river that has all 4 types! In just a few miles too.
At various points on that river, I can not paddle and the river moves me pretty good, I will basically sit in one place but get out of the canoe and the river tries to sweep my legs away, some spots it's like being in a swimming pool and seems no water is moving. Another place I can sit something on the bottom of the river and it stays there, but the river is pushing everything on top hard.
What type of river location you have as I mention above would be a major factor how deep you need the wheel. If it's almost all surface water moving then you need to be shallow, since going deeper into standing water will just add drag.
If you have still water on top and fast currant deeper, then you need to get past the still water and into that deeper currant.
"How big should the paddles be in relation to the diameter of the wheel, also, how much of the wheel should be submerged? Any other tips on making this project work?"
Now as far as the size of the paddles, I don't actually think it matters as far as wheel size, what you need is to capture as much force from the water as you can, where ever it is, and with out any extra drag in the slower or standing water.
So lets say the top water is fairly still and you have strong currants 2' deep. You need a larger wheel so you can reach 2'deep and then paddles at the 2' mark and deeper where the currant is. So lets say you only have 1' of strong currant. Ok being it starts at 2', then you need 3' of wheel in the water and the paddles would be 1' high so as to catch the currant which is only at 2' to 3' depth. Larger paddle would create drag in still water above the 2' mark (plus everywhere else too) and provide no bennifit.
On the other hand if it's only top water with a currant then you need less wheel in the water, only as deep as the currant goes and a paddle that reaches from the top to bottom of the currant area, but without extending into still water.
If the entire river is moving strong top to bottom then you could use a wheel and paddle that reaches to nearly the bottom if you wanted.
Your just dead in the water if it's still like a swimming pool and not much is gonna help.
As far as how wide the wheel and paddles should be side to side, that depends on the force of the currant and how much you need to capture. If a mild currant and you need alot, then of course you need wider blades to catch more moving water. If a really strong currant and you only need a little power you can use narrower paddles.
How far the wheel needs to extend into the water is only a factor of where the water is moving best.
You might set the same wheel up on 3 rivers and have to make changes each time to get the best results, or maybe even any results at all.
If you ever fished much in rivers you might know alot of what I mean about depth and currant.
Cast a bobin into the river with a lure on 1' of line and it just sits in one place, put the lure on a 3' line cast to the same spot, bobin now tries to run down river hard! That's still water on top and curant near 3' deep.
Sometime it's oposite, bobin and 1'lure tries to run away, but a 3' lure just sits there. That's top currant and still wter below.
Might be a good idea to take a pole, some bobins, lures, and sinkers to the river with you as currant checking tools 
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