They make both wall systems and flooring systems for wet basements. Sorry I am not posting links but don't feel like shopping today. A friend just put up wet basement panels (google that) and they look good and they are waterproof.
Whether this is the right solution for you depends on your house and how long you expect to be there. If your yard is graded poorly, your basement walls are in really bad shape, or there is some other intrinsic issue that will always cause moisture I would personally not do what Chris or bib suggested.
First it is a basement. How much are you willing to spend to make a place that naturally has water nice - there are "newer houses" and houses built in dry areas that don't have issues which are perfect to renovate - but that is not your house. I have seen many disappointed people spend thousands of dollars waterproofing basements to have their (newly finished) basement spring a leak the next year.
My suggestion is make the basement livable, safe, and enjoyable. Knowing that it could get flooded at some point. I would suggest putting in water-proof (rubber) planked flooring - HD sells this. Even if it is soaked it can be cleaned and reused. I would suggest some waterproof panels set about 4-6 inches from exterior walls. I would suggest no insulation other than rigid foam paneling. And if you do the foam paneling I would allow space for it to breathe. I would suggest neither drop ceiling or drywall for the ceiling - just paint it black or white (white you have to clean more).
If this extra space is really important to you - and you want it nice - I would suggest that you look at adding an addition to your house. After you do all of that waterproofing, foundation repairs, and finish your basement your $$$ is getting pretty close to adding on plus you don't have to worry about it being flooded.
This is going to be sort of a comment, or at least not a complete answer.
When designing the system, the goal is not to create a breeze through the space. The goal is to control pressure equalization in such a way that the air is mixed and dehumidified (conditioned).
Having one large return at one end of the space, and a large return at the other may work. But it's not going to be as efficient, and will require maintaining an open area for which the air to flow (no closed doors).
Let's back up a bit, and think about how the system works...
The fan (or blower) is going to create a high pressure area "in front" of it, which causes the pressure in the supply ducts to increase. The higher pressure air flows from the supply ducts, in an attempt to equalize the pressure between the supply ducts and the rooms. At the same time, the fan (blower) has also created a low pressure zone "behind" it. The higher pressure air in the rooms is forced into the return ducts, in an effort to equalize the pressure between the rooms and the return ducts.
So the system works by forcing conditioned air into the rooms, and having unconditioned air pushed into the system.
Wherever you have a supply duct, you want to have and unobstructed return duct. Otherwise you'll end up with a high pressure area in one place, and a low pressure area in another.
In your example, if you have a single supply duct feeding the kitchen, and a single return in the bedroom. What happens if the bedroom door is closed? In fact, the bedroom door in your drawing would have a tendency to be pushed closed, if you installed the system the way you've described. You'll be creating a high pressure zone in the kitchen and living room, and a low pressure area in the bedroom. Which means the door could be closed by the pressure differences.
You'll want to install a supply and return in each separate area. In your case that would mean installing a supply and return in the living room/kitchen, bedroom, and storage area.
Unfortunately, duct design is a bit of science, and a bit of art. Covering all the information you'd need to know, would be nearly impossible in this short answer format. In my opinion, duct design is not a job for an amateur. At the very least, you'll want a pro to assist in that aspect of the job.
I know I've rambled a bit, and I'm sure you don't want somebody to tell you not to do it yourself. But in my opinion, it's better to have it done right, then to have it done cheap. Aside from all that I've mentioned. It's impossible for somebody to design a duct system without knowing the equipment you're going to use, or the actual dimensions of the space. So there's no way anybody on this site could provide a duct design for you, with the information you've provided.
My advice... Get an HVAC contractor to design the system for you, then you can do the work of installing it yourself.
Best Answer
I would stain the concrete or tile it and leave the drains there. I don't think carpet belongs in basements. You are just asking for trouble like mold, mildew, etc.
Good luck!