I just noticed this thread in my monthly mail. Bravo, and apologies for lateness/duplication.
While the air under the car is certainly in shear, a popular way to get around the lack of a moving ground plane is to simply mirror the shape and observe the gap between the two "bottoms." An ideal shape in free air, when brought close to its twin, will distort progressively so that the two adjacent surfaces become flat as they touch, leaving no visible seam as they merge into one body of the original shape but double volume.
The exposed wheel bottoms add frontal area, but enough ground clearance for easy street use is good, because it can better handle the inevitable imprecision. Wheel exposures benefit from fairing, but make sure to align it with the local airflow, not just the wheel itself. Use smooth wheels and wheelwells, with minimal gaps at the openings. Detailing matters quite a bit. Don't let air leak from one pressure zone to another through the interior, except through planned, tapered ducts.
For your cockpit, you want to create a stable, attached vortex to produce a virtual roof. Some decent examples can be found in the LeMans sports classes from the 60s, before the lads discovered downforce. Draw a shape that has a fairly flat roof the shape of your cockpit opening, and then just cut a hole. The ideal shape for the cavity is very rounded for easy, fast circulation, blended to a rounded rear edge that won't catch air, and is lowered enough to account for the growth in the boundary layer. Anything you can do to feed high-velocity air to the start of the gap will help reduce turbulence. There isn't much in the literature, but I could maybe scan what I have if there's interest.
Exposed roll bars are very nasty. Just strategic roughness helps if it has to stay round, but even a crude cardboard or foam fairing taped on can hardly miss at halving the drag. Good ones can reduce it by an order of magnitude.
The softer the ride, the less momentum lost on each bump, and the lighter the tires you can use. You can achieve suspension just by controlling the flex in the frame. The most basic improvement would be to have the batteries and/or seat on springboards. With more travel comes more potential troubles about keeping things perfectly aligned and not scrubbing, but it is easy to make big gains. It is pretty easy to design A-arms that replace the hinge with a wide quarter-elliptic spring, and then blend in to the rest of the structure without heavy transitions. I have articles on full suspension from frame flex, and on Coroplast fairing construction linked at
The Car Cycle
Re: Selection of materials, from among those popular in the field.
Choose the ones you enjoy working with, and have the tools/facilities for. You can machine aluminum with carbide woodworking equipment. The exotics are wasted if you don't have craftsmanship. Honeycomb panels can be handy and light. For a one-off, I'm partial to using a lot of wood, and with epoxy encapsulation it can be excellent structurally, but the coating must be protected from even small nicks where water can enter, and from UV light. A thin layer of fiberglass gives good mechanical protection, and will tie together many wooden members.