Is there really a direction of time? Laws of physics are (almost) time symmetric, so a time-reversed description of a physical process is as qualified as the original one. What's the reason then, that in reality one version seems to prevail but not the other?
Entropy is smaller closer to the big bang and larger on the other side, so it's asymmetric with respect to time. But asymmetry is not direction. "Entropy increases from big bang forward" and the time-reversed version "Entropy decreases towards future big bang" are equally good descriptions of this asymmetry. Are we choosing the former over the latter? If so, mustn't there be some reasons other than reversible laws and boundary conditions that legitimize our choice? What might they be? 
Everything we remember happened when entropy was lower, not when it is higher. Could this be one of the reasons why we choose one over the other?    
 A: This is a question that cannot be answered without understanding what time is. And I think the best we can say is that time is a quantity we use to compare events. But the compared feature is the ordering. Bearing that in mind, we cannot say much about its reality, even less about its direction. 
In other words, despite the fact that you can order events, and even talk about their duration, we cannot say much about the fundamental nature of time.
Now there is a diference between physical laws and the quantities they relate, as time in this case. Physical laws that are time symmetric simply summarize observed phenomena, and the terms they use show no preference for time. But this just means in my opinion, that the initial conditions are the ones carrying information about the succession of states in the system. Thus in a sense, the are symmetric because the very processes  governing change are time symmetric; but this just means that the physical process governing change is independent of the starting point, or initial conditions. Then the physical laws do not say by themselves whether time is a property of the whole universe, or the interacting bodies, or if is just a concept with no physical meaning. Even less about whether it has or not a preferential direction.
A: Time is just a scale we use to measure the rate of processes or to measure the interval between 2 events.  Time is not a stand-alone entity and it does not exist alone independently.  All the means (like clocks) we employ to measure Time use some standard physical changes as their fundamental measuring units of Time.  So Time has no direction of its own, like a tape for measuring distance has no direction of its own, even if we use that measuring tape to measure, for instance, the length of one-way street.  The process on that street (one-way traffic) does not mean that the measuring tape really runs then same direction as the traffic.  It's the same with Time.
A: 
Everything we remember happened when entropy was lower, not when it is higher.

This is not even wrong. The entropy of any given system (and we get to choose the definition of the system) can decrease as long as it is not a closed system.
So, what system are you talking about? You don't know. We don't know. That is why this question doesn't make sense to ask.

Laws of physics are (almost) time symmetric, so a time-reversed description of a physical process is as qualified as the original one. What's the reason then, that in reality one version seems to prevail but not the other?

What do you mean one version? Again, this is pretty close to non-sense so it is hard to even parse what this question could mean.
If you are asking how the direction of time is related to physical processes and the increase in entropy, the typical answer is via Boltzmann's H-Theorem. (N.b., this theorem was proven before time-reversal symmetry breaking processes were known, but nevertheless it still holds.)
Boltzmann's "H-Theorem" says that
$$
H = \sum_i p_i \log(p_i)\;,
$$
never increases due to state transitions and the lowest it gets is in equilibrium (where its value depends on what quantities are conserved for the system under consideration).
Boltzmann's "H-Theorem" can be demonstrated by using:
$$
\frac{dp_i}{dt} = \sum_j \Gamma_{i\to j}p_i - \Gamma_{j\to i}p_j\;,
$$
along with the fact that
$$
x \log(y/x) < y - x\;.
$$
