Why are transitions and emissions in HMM assumed to be independent? In the hidden Markov model we use two matrices. The first one, called the transition matrix, determines probabilities of transitions from one hidden state to another one (the next one). The second matrix, called the emission matrix, determines probabilities of observations given a hidden state.
In other words, we can imagine a system as being in a state (which is hidden, unobservable) and this hidden state determines the probability of the next hidden state as well as probability of a given observation.
It means that we assume that a "jump" (or transition) to the next hidden state and "generation" of a certain observation are independent events. Is this a limitation of the model? Or, in case of such a dependency, is there a way to reformulate the model (for example, by redefining states) such that these two "steps" (transitions and emissions) are independent?
 A: Yes, it's a limitation of the model.
In most dynamical systems that you might care to think about using an HMM, the act of measurement or observation is usually conceived and/or constructed in such a way that it is assumed not to affect the system under observation.  We want to imagine the observer as being passive and completely independent from the system, and then make inferences about how the hidden states would evolve, even if the observer were not present.
An example might help to make this a bit more concrete.  Imagine that you are tracking a satellite moving through the sky with a telescope.  The "hidden states" in this case would be the position and velocity of the satellite at any given moment in time.  The "transition process" is governed by whatever forces are acting on the satellite; certainly gravity at all times, but also possibly other things such as the thrusters in a reaction control system.  The "observation" at any given moment consists of two angles, the altitude and azimuth with respect to the horizon.  Now ask yourself, how weird would it be if the very act of observing the satellite through a telescope, hundreds or even thousands of miles away, could actually cause it to speed up or slow down (thereby impacting its hidden state) spontaneously?
Lots of dynamical estimate problems are like the satellite and telescope problem: the act of observation itself doesn't affect the hidden states very much, or at all.  HMMs, or similar/related concepts, are a perfectly valid and useful way of thinking about these kinds of systems.  The independence of the observation and the state transition are a desirable feature of the model.
It's worth noting that there are systems where the act of observation is presumed to affect the thing under observation.  This kind of interdependence is actually an integral feature of quantum mechanics.  The term of art in quantum mechanics for the moment when an observation actually affects the thing under observation is wave function collapse.  To model a quantum mechanical dynamical system in a way that would be analogous to how we use HMMs, you would indeed need to assume that the observation and the state transition are coupled events.  It's worth noting that if you google quantum filtering or quantum Markov filter, there are several hits, indicating that it does appear to be an active topic of current research.
A: In wikipedia article it is stated that next hidden state  $x(t+1)$ conditionally depends from previous hidden state  $x(t)$:

But we can't say that observation $y(t)$ and next hidden state $x(t+1)$ are independant. Because if you observe  $y(t)$ then you have some knowledge about $x(t)$ probability distribution and then knowledge about $x(t+1)$ distribution. 
For example, if you have that $y(t)$ just equals hidden state: $y(t) = x(t)$ then knowing $y(t)$ value gives you $x(t)$ value. And then you can derive $x(t+1)$ distribution from $x(t)$ value.
But you can say that $y(t)$ are independant with  $x(t+1)$ given  $x(t)$. If  $x(t)$ is fixed (you don't know the exact value, you just know it is fixed) then $y(t)$ observation gives you nothing about $x(t+1)$.
