Intuitively Understanding Work and Energy
You may want to see Why does kinetic energy increase quadratically, not linearly, with speed? as well, it's quite related.
Mainly the answer to your questions is "it just is". Sort of.
What is a definition of energy that doesn't use this circular logic?
Let's look at Newton's second law: $\vec F=\frac{d\vec p}{dt}$. Multiplying(d0t product) both sides by $d\vec s$, we get $\vec F\cdot d\vec s=\frac{d\vec p}{dt}\cdot d\vec s $
$$\therefore \vec F\cdot d\vec s=\frac{d\vec s}{dt}\cdot d\vec p$$ $$\therefore \vec F\cdot d\vec s=m\vec v\cdot d\vec v$$ $$\therefore \int \vec F\cdot d\vec s=\int m\vec v\cdot d\vec v$$ $$\therefore \int\vec F\cdot d\vec s=\frac12 mv^2 +C$$
This is where you define the left hand side as work, and the right hand side (sans the C) as kinetic energy. So the logic seems circular, but the truth of it is that the two are defined simultaneously.
How is kinetic energy different from momentum?
It's just a different conserved quantity, that's all. Momentum is conserved as long as there are no external forces, kinetic energy is conserves as long as there is no work being done.
Generally it's better to look at these two as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.
Why does energy change according to $Fd$ and not $Ft$?
See answer to first question. "It just happens to be", is one way of looking at it.
What is a definition of energy that doesn't use this circular logic?
Historically, people had no clue that energy was conserved, basically because it's not obvious that when mechanical energy appears to dissipate at least partially into nothingness, actually it's turning into heat. Often the temperature changes involved are very small and not noticeable. But people had a clear intuitive idea that $Fd$ was a good figure of merit for what was being done by a horse or a steam engine, so they called it work. Later, when conservation of energy was discovered, they had this preexisting numerical scale, and they realized that it was a measure of the transfer or transformation of energy, so they started using it as the unit of energy.
From a modern point of view, there is another, nicer way to proceed. We start with some more fundamental definition for energy. For example, we can define some standard form of energy such as kinetic energy. Then, exploiting and constrained by conservation of energy, we determine a numerical scale for this form of energy and for other forms of energy that can be converted to and from it, such as gravitational potential energy. The Feynman quote in TreyK's answer is a presentation of this philosophy. One can then define work in terms of energy, as the amount of energy transferred by a macroscopic force, and prove a theorem that it's measured by $W=Fd$ under certain conditions. Or we can stick with $W=Fd$ as a definition of work, in which case we can prove as a theorem that it equals the energy transferred.
[...] $E_k=\frac{1}{2}mv^2$, but why is that?
The factor of 1/2 in front is purely a historical artifact. Conservation laws don't change their validity when you change units, so we could have any factor out in front that we liked. But if, for example, we chose to define kinetic energy as $mv^2$, then we'd have to change the numerical factors in every other equation relating to energy, e.g., we'd have $W=2Fd$.
The proportionality to $m$ has to be that way because conservation laws are additive. E.g., if KE was defined as $m^2v^2$, it wouldn't be additive when you added the energies of two different objects.
The factor of $v^2$ doesn't have to be that way logically, and in fact it isn't really $v^2$ -- relativistically the correct equation is different, and $v^2$ is only an approximation for velocities that are small compared to the speed of light. However, if we assume Newtonian mechanics to be a good approximation, then it does have to be $v^2$. There are various ways of proving this. For example, in Newtonian mechanics momentum equals $mv$ and is conserved. If you take KE to be proportional to $v^2$ and also want energy to be conserved regardless of your frame of reference, then you get a condition that is exactly the conservation of $mv$. For any other proportionality besides $v^2$, the behavior of the conservation laws for energy and momentum would not be consistent with each other when you changed frames of reference.
After more digging, I came up with this quote from Feynman -
There is a fact, or if you wish, a law governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy.
It states that there is a certain quantity, which we call “energy,” that does not change in the manifold changes that nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says there is a numerical quantity which does not change when something happens.
It is not a description of a mechanism, or anything concrete; it is a strange fact that when we calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.
It is important to realize that in physics today, we have no knowledge of what energy “is.” We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. It is an abstract thing in that it does not tell us the mechanism or the reason for the various formulas.
As Manishearth's answer demonstrated, it is certainly possible to show the mathematical principles that go into understanding energy, but it seems to me to be a formula meant for mathematical convenience (as is Torricelli's equation), and not something meant to be intuitively understood in and of itself -
Generally it's better to look at [kinetic energy and momentum] as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.