Why are position and velocity enough for prediction and acceleration is unnecessary? In classical mechanics, if you take a snapshot and get the momentary positions and velocities of all particles in a system, you can derive all past and future paths of the particles. It doesn't seem obvious why the position and its first derivative are enough and no further derivatives are needed.
For some reason the accelerations (forces) can be expressed by formulas that only mention the position and velocity of particles. For example, the gravitational force only requires knowing positions but some electromagnetic things need velocities as well. Why doesn't anything need the second derivative (acceleration)?
Does this say something about the universe or rather about our way of analysis?
Could we come up with a theory that only requires a snapshot of the positions? Could we devise a set of concepts and formulas where the second derivative is also required for prediction and instead of forces we'd be talking about stuff that induces third derivatives of motion?
Does modern physics (e.g. relativity) have something to say about this curious thing?
 A: The reason that you only need to specify initial position and velocity to exactly solve the equations of motion for a system is simply because Newton's Second Law (which is the equation governing motion in Classical Mechanics) is a second-order differential equation. The upshot is that to solve a 2nd-order ODE, you basically need to take 2 integrals. Each integral will have exactly one undetermined constant of integration, so by specifying those numbers with your initial conditions, you have uniquely specified your problem's solution.
A: Note: I'm not a professional physicist. Below notes are accumulated from resources I found trying to chase this question.
First let's re-phrase the question: Are there any fundamental force law that depends on time derivatives of position of order higher than 2? The related terminology in academia is higher order gravity. 
View 1: Higher order Lagrange equation yields very short lived universe
This paper titled Higher Derivative Theories of Gravity has a very detailed math concluding that higher order gravity would not be compatible with our universe. Another paper Higher-derivative Lagrangians, nonlocality, problems, and solutions also has more technical details on failures of using higher order derivatives to generalize General Relativity further. This Quora thread, Research Gate thread as well as this answer has assertion that system with differential equations of higher order causes Ostrogradsky instability and hence cannot exist. Note that few people in Quora thread have challenged that higher order natural forces cannot exist by pointing out that Radiation Reaction has force that depends on 3rd derivative of electron position. There are also systems such as Euler–Bernoulli beam and Korteweg–de Vries which indeed have higher order derivatives. Unfortunately I don't see any arguments to counter them. 
View 2: Universal Forwards Causation doesn't allows higher order derivatives
In this published paper Why Physics Uses Second Derivatives author lays out the following argument:

Thus if there is a fundamental force law, which operates by setting the
  second derivative of position, then the first derivative of position
  that is causally relevant (velocity) must be a past derivative, while
  the second derivative (acceleration) must be a future derivative.
  However, if there were a fundamental ‘yank’8 law, which operated by
  setting the third derivative of position, then the first two causally
  relevant derivatives of position (velocity and acceleration) would
  have to be past derivatives. So if there were both a fundamental force
  law and a fundamental yank law, then acceleration would both have to
  be a past derivative and a future derivative, which is a
  contradiction.

View 3: Higher order derivatives violated SEP
Another interesting point of view is provided by this answer which states that (per my understanding) if there was an universe where existed natural forces of higher order derivatives then observer in those frame of references will violate Strong Equivalence Principal. This is seconded by this paper on Higher order gravities and the Strong Equivalence Principle as well.
Side Note
Apparently there exist fractional calculus where you have fractional derivatives! So the related question would be: Are there any natural forces that depend on fraction time derivative? There is some analysis to this question in this paper.
A: 
 Why doesn't anything need the second derivative (acceleration)?

Only Newton's gravity law does not use acceleration in the expression for force. In electromagnetic theory with retarded fields, forces are functions of past positions, velocities and accelerations of the charged particles.
A: I am trying to answer your questions one by one. By the way, I saw your comment.
1. You say in the question: "Why doesn't anything need the second derivative (acceleration)?"
Yes the acceleration is needed for obtaining the velocity. I some situation we are given the force and the mass, as in an electrical or gravitational field, not the velocities. Then the velocities may be calculated,
$$v = \int_{t-0}^{t_1} a(t) dt.$$
2. You ask whether "we decided to conceptualize things that generate second derivatives (and named them forces). But maybe it's an actual property of the universe that it's only "2 derivatives deep". "
It's history here, beginning with Newton and Keppler. The former gave us the 2nd principle of mechanics, and the latter, the orbital movement of the celestial bodies. After them, we built our new theories using the concept that they introduced/used.
3. "Could we devise a set of concepts and formulas where the second derivative is also required for prediction and instead of forces we'd be talking about stuff that induces third derivatives of motion?"
A law of movement doesn't have, in principle, to lead to a 2nd order equation. It may lead even to a 3rd order if someone gives you $da/dt$. I don't know such practical cases, but the future is ahead.
4. "Could we come up with a theory that only requires a snapshot of the positions?" 
A law of motion is $\vec X(t)$. In some cases it is preferable $\vec X(s)$, where s is some parameter, but only a collection of positions won't help. 
A: because we want to find the trajectory of the particle and if we know it's position in different times we can calculate it's motion. Also, the acceleration can be obtained when we have the X(t). then acceleration is not a new information to use and it is embedded in the X(t) (acceleration=$d^{2}X/dt^{2}$)
A: "Could we come up with a theory that only requires a snapshot of the positions? ...Does modern physics (e.g. relativity) have something to say about this curious thing?"
Yes. The Theory of Special Relativity posits that there is no favored rest frame, and therefore there is no such thing as absolute velocity. In other words, velocity can only be measured relative to another object, but not against any "0" velocity, which does not exist.
As a consequence of this, in any "inertial" (non-accelerating) frame of reference[*], all laws of physics must still be valid, independent of velocity. One consequence of this is that the fundamental constant, $c$, the speed of light in a vacuum, must be identical in all such reference frames, regardless of how fast you are traveling relative to any other object (this is what leads to length contraction and time dilation). Another consequence of this is that Maxwell's equations have to be reformulated, using Tensor Calculus, into a form that is invariant under Lorentz Transformations.
Essentially, what appears in one reference frame as a magnetic force (due to a moving charge), will appear as an electric force in a reference frame that is moving with charge (i.e. a frame in which the particle appears stationary).
[*] For accelerating frames of reference, General Relativity is required.
