For the sorts of vehicles we're used to, like cars and aeroplanes, there are two contributions to drag. There's the drag caused by turbulence, and the drag caused by the effort of pushing the air out of the way. The streamlining in cars and aeroplanes is designed to reduce the drag due to turbulence. The effort of pushing the air out of the way is basically down to the cross-sectional area of whatever is pushing its way through the air.
Turbulence requires energy transfer between gas molecules, so you can't get turbulence on length scales shorter than the mean free path of the gas molecules. The Wikipedia article on mean free paths helpfully lists values of the mean free path for the sort of gas densities you get in space. The gas density is very variable, ranging from $10^6$ molecules per $\mathrm{cm}^3$ in nebulae to (much) less than one molecule per $\mathrm{cm}^3$ in intergalactic space, but if we take the value of $10^4$ in the table on Wikipedia the mean free path is $100\,000\ \mathrm{km}$. So unless your spaceship is very big indeed we can ignore drag due to turbulence.
A sidenote: turbulence is extremely important in nebulae, and a quick glance at any of the Hubble pictures of nebulae shows turbulent motion. However the length scale of the turbulence is of the order of light-years, so it's nothing to worry a spaceship.
So your spaceship designer doesn't have to worry about the sort of streamlining used in aeroplanes, but what about the drag due to hitting gas molecules? Let's start with a non-relativistic calculation, say at $0.5c$, and use the density of $10^4\ \mathrm{cm^{-3}}$ I mentioned above, and let's suppose that the gas is atomic hydrogen. If the mass per cubic metre is $\rho$ and you're travelling at a speed $v$ then the mass you hit per second is:
$$ m = \rho v $$
Suppose when you hit the gas molecules you accelerate them to match your speed, then the rate of change of momentum is this mass times your speed, $v$, and the rate of change of momentum is just the force so:
$$ F = \rho v^2 $$
An atom density of $10^4\ \mathrm{cm^{-3}}$ is $10^{10}\ \mathrm{m^{-3}}$ or about $1.7 \times 10^{-17}\ \mathrm{kg/m^3}$ and $0.5c$ is $1.5 \times 10^8\ \mathrm{m/s}$ so $F$ is about $0.4\ \mathrm{N/m^2}$.
So unless your spaceship is very big the drag from hitting atoms is insignificant as well, so not only do you not worry about streamlining, you don't have to worry about the cross-section either. However so far I've only talked about non-relativistic speeds, and at relativistic speeds you get two effects:
- the gas density goes up due to Lorentz contraction
- the relativistic mass of the hydrogen atoms goes up so it gets increasingly harder to accelerate them to match your speed
These two effects add a factor of $\gamma^2$ to the equation for the force:
$$ F = \rho v^2 \gamma^2 $$
so if you take $v = 0.999c$ then you get $F$ is about $7.5\ \mathrm{N/m^2}$, which is still pretty small. However $\gamma$ increases without limit as you approach the speed of light so eventually the drag will be enough to stop you accelerating any more.
Incidentally, if you have a friendly university library to hand have a look at Powell, C. (1975) Heating and Drag at Relativistic Speeds. J. British Interplanetary Soc., 28, 546–552. Annoyingly, I have Googled in vain for an online copy.
The first factor in stability is pitch inertia. Even though it is just a sheet of paper, its moment of inertia around the pitch axis is quite high, so any pitch motion is slow.
Next is aerodynamic pitch damping: As the paper starts to pitch, local forces at the edges will produce a stabilizing moment.
That the paper will slowly pitch up is due to the location of the center of lift, which will act at a quarter of the chord. The precise location is slightly ahead of the quarter chord, the more so, the more slender the paper is (ratio of span to chord length). This creates a pitch-up moment, and since the paper itself is rather floppy, will bend the whole paper into a graceful curve. This again helps to stabilize the paper, because a negatively cambered wing is naturally stable.
Your paperclip weight is needed to shift the center of gravity forward to one quarter of the paper airplane's chord. This is where the resulting lift force will act, so by shifting the cg forward, you minimize pitch moments. You could as well fold the first half of the sheet of paper into a tight roll of paper - this will produce the same effect as the paperclip. Once the pitch angle increases, so does lift and angle of attack. Beyond an angle of attack of maybe 10° or 15°, airflow on the top will separate, which will shift the center of lift backwards. Also, lift stops to increase with further angle of attack increases. Now the lift creates a pitch-down moment, which helps to regain the attitude at which the flow is still partially attached. Especially when using stiffer cardboard, this stabilizing effect of flying with a partially stalled wing is easy to reach once the center of gravity is right.
In folded paper airplanes, you will notice that a slightly bent-up trailing edge will help to stabilize the paper airplane, whereas folding it down will make sure it goes into a dive quickly. By folding the trailing edge at least partially up, you will create a local area of lower lift (or actual downforce), which will see a proportionally stronger increase of lift when the whole paper airplane pitches up. This results in a pitch-down moment, and vice versa. This is all what is needed to give a paper airplane positive stability.
A simple sheet will be too floppy to benefit from the stabilizing effect of a bent-up trailing edge, though.
Best Answer
You want your parachute not to be a parachute, but a wing. The difference is that it has horizontal velocity, and the air flows smoothly over the top and bottom surfaces.
In addition, you want to minimize drag, because sink rate is proportional to drag. The main way to minimize drag is to minimize speed. So it needs forward speed, but no more than necessary.
Check out paragliders, because that's what they do. I've seen these things in action. They stay up a long time.