Why is an Alcubierre drive shaped like it is When I look up the Alcubierre drive I keep seeing that the drive powering the bubble is shaped like a circle - is this necessary for the creating of the "warp bubble" that travels forward? Does the theory say anything about what happens if you do not shape it like a circle? Is it possible to have it warp spacetime in one 'direction' (i.e. it contracts space in one direction (--->) but does not expand space in the other direction (<---))? Or will there always be created a uniform bubble around whatever exotic matter is used for the drive?
 A: Einstein's equation is:
$$ G_{\alpha\beta} = 8\pi T_{\alpha\beta} $$
In this equation $G_{\alpha\beta}$ describes the curvature of spacetime and $T_{\alpha\beta}$ describes the mass/energy distribution, so the equation relates curvature to mass/energy. Solving this equation is the basic task in general relativity.
There are two ways to look at the equation. The most obvious is to start with some known distribution of mass/energy $T_{\alpha\beta}$ and solve the equation to get the spacetime geometry. This is how Schwarzschild worked out the geometry of a black hole, and Friedmann et al worked out the geometry of an expanding universe. However you can reverse the process i.e. start by deciding what geometry you want and then work out the corresponding mass/energy distribution $T_{\alpha\beta}$. This is what Alcubierre did.
Alcubierre started with the spacetime geometry he needed for his FTL drive, then solved for the corresponding mass/energy distribution. The result turned out to be a ring of exotic matter, which is why the Alcubierre drive uses a ring - it wasn't designed that way in advance, that's just the way it worked out.
You ask what happens if we use different shapes, but that question is far easier to ask than to answer. Calculating $T_{\alpha\beta}$ from a known $G_{\alpha\beta}$, as Alcubierre did, isn't that hard to do. However the more conventional calculation i.e. starting with $T_{\alpha\beta}$ and calculating $G_{\alpha\beta}$ is a formidable problem, and indeed can only be done precisely for a few special cases. In the vast majority of cases we have to resort to numerical methods and an extremely large computer.
So there isn't a simple answer to what happens when we change the shape of the ring. Very small changes could probably be tackled using a perturbative approach, but in general you'd need to reach for your (super)computer.
A: Since you don't locally travel faster than light it is hard to call what you are doing FTL if you don't contract on one side and expand on the other.
That said, you don't have to contract space in front of you. The Natario drive is claimed to pancake the space in front of you, thus reducing the distance to your destination but without contracting space in front. (Thank you to diffeomorphism for correcting my poor citation.)
For instance if you are in a big crunch scenario with a globally uniform contracting spacetime then it hardly makes sense to call sitting around not moving being a warp drive even if the comoving proper distance between you and a distant place is decreasing at FTL speeds.
It doesn't require exotic matter so doesn't seem exotic.
What is the difference between contracting and pancaking?
Contract is a standard term meant to oppose expand. Like when the universe expands or contracts. They give a definition of contracting/expanding or lack thereof in the Natario paper but I'm not sure how physically meaningful the distinction is, which is why I merely said they claim. 
The term pancake is my own. If you'd rather you could pretend I said the region in front of you contracts in the forward direction and expands in the sidewise directions. (But all of that is happening in the region in front of you.) So the idea is that you aren't destroying space in front of you in any sense.  So I imagine a sphere in front of you becoming a pancake just like squishing a ball of dough. Similarly you can set it up to not create space behind you. Then connect them all up.
The Alcubierre drive sort of piles things up in front of you which is quite destructive for anything in front of you if you managed to slow down because you have lots of space debris you've carried with you that shoots off.
The Natario drive might be safer for the people in front of you when you break if it acts more like a two sided snow plow and let's thing go around you rather than pile up in front of you. 
But I'm not sure if it safer by an important amount, I haven't seen anyone write a paper on that.
