Hitting a tire with a sledgehammer - how to produce the most force? Assume I have to hit a tire that's lying flat on the ground
I have two options
1. Stand towards the tire with the hammer on the ground behind me, from which point I swing it in a partial circular motion towards the tire (refer to painting - start at point B).
2. Start with the hammer above and a bit behind my head then drive it down in a half circular shape (refer to painting - start at point A).

The way I see it, the acceleration I amass up until the last moment in which the tangential acceleration is vertical to the direction in which I want to accelerate the hammer is irrelevant and contributes nothing to the production of force against the tire.
Therefore, according to my understanding, the motion between point B and point A is meaningless except that it makes me exert force.
Did I understand it correctly?
 A: Force is given by Newton's second law: $F = dp/dt$. What matters is the momentum of the sledgehammer just before it hits the tire, and the momentum of the sledgehammer immediately afterwards.
The acceleration you amass (this terminology is loose, however) before the sledgehammer hits the tire goes towards increasing the sledgehammer's momentum. Since the momentum of the sledgehammer just before it hits the tire is greater, the amount of force exerted on the tire also increases. If you're able to impart to the sledgehammer the same amount of momentum by starting from point A as opposed to point B, then you indeed don't need to start from point B.
A: You need to lift the hammer to get it to point A.  That requires a certain amount of energy regardless of the direction in which you're lifting it (e.g. from the tire to point A or from point B to point A)  If you provide just enough energy to get it to point A so it is motionless at point A, no difference.  But it you provide more energy than necessary, you're better off lifting from point B because that extra energy will go into rotational energy and will be added to whatever you can provide in moving the hammer from point A to the tire. Otherwise, you will need to slow the hammer and reverse its direction at point A: you will waste that extra energy.
A: If you start from zero velocity at point A and you put all your force into it, that's as much momentum as you can get starting at point A. Some of what you do will be pulling back on it to change the direction of force.
If you can start at point B and get a positive momentum when you reach point A, and if you can still put all your force into it from point A on, just as you did starting at point A, then you get more momentum at the end.
If it takes so much out of you going from point B to point A that you can't do as much from point A on, then it depends. 
The issue is not the physics. The issue is the abilities of human muscle. If instead of your muscles you had a machine that delivered the same amount of torque no matter how fast the weight was already moving, and it could deliver enough against gravity that you already had some momentum when you got to point A, then definitely you'd get more that way. If your machine couldn't deliver enough torque to get from B to A, then it would have to start at A.
If it can apply more force at high speed, so the major part of the momentum comes near the end when gravity has been working for you, then a little extra velocity at point A wouldn't help much. 
Again, it's about the limitations of your ability to apply force. Different rules for that give different results.
A: The bigger the arc that you use your constant force on, the faster the hammer head will move, a = F over mass and v= 0.5 at^2, note the t squared part which means that you can get a lot of energy in the head. This assumes as others have mentioned that you are not having too much trouble with lifting the hammer from B to A. (maybe try a sideways stroke?) At impact the force that your are exerting is small compared to all the built up energy or momentum during your arc that is in the hammer head.  If the tire is very elastic the collision is more elastic and as the tire has no where to go the hammer head dangerously bounces back, be careful.  If the tire is more deformable (softer?) the tire can absorb some of the energy and the head does not bounce back as far. 
A: If the hammer were massless, then you would be correct.  The only force on the tire would be that created by the torque from your muscles.  As you could create almost the same torque with the hammer resting against the tire, there is no need to move it.
However, since the hammer has mass, we can increase the force on the tire by making it accelerate the mass ($F = ma$).   So anything that increases the speed of the hammer prior to contact with the tire should help.  Now you have two reasons to lift the hammer.
The first is simply biomechanics.  Let's imagine that we are attempting this exercise with no gravitational acceleration present (and our hammer wielder is appropriately bolted to the floor).  It's entirely possible that the person can generate more speed by taking a longer swing.  Yes, this means that a greater acceleration is required.  But different stances and swings might be able to generate more efficient forces (perhaps by allowing additional muscle groups to be used).  Analyzing the biomechanics in this case could be complex.
The second reason is that this is happening in a gravitational field.  So much of the work raising the hammer to pointA is used to increase its gravitational potential energy.  That energy is converted into kinetic energy as it falls toward the tire.  You could simply drop the hammer from a height above the tire without exerting any additional force and you'd still create an impact just from the gravitational acceleration.  Pulling on in on the way down just adds to the energy and subsequent impact.
A: This question is more about bio-mechanics than physics. If you could swing from B without getting tired and/or putting your body in a less optimal position to swing from when you get to A, then it is better to start from B. In practice, the muscles in your abs and shoulders will probably be harder to utilize optimally if you start from B, you will likely not be flexible enough to keep your feet planted or to provide force throughout the entire motion. 
I suggest you take a look at golf swings or lumberjacks. Notice a nice long wind-up is important in both cases. But they don't wind up so much that they end up in a position to compromise their stability so that they can apply maximal force coherently with their muscles.
In other words, if you were to design a machine to do this, it's better to start from B. For a human, flexibility and stability will likely matter much more than simply having a longer path length across which you can apply force.
