How to interpret irreversibility in time? I'll quote Feynman's Lectures, chapter 52 (Symmetry in Physical Laws) of volume 1:

[...] If we see the egg splattering on the sidewalk and the shell cracking open, and so on, then we will surely say, "That is irreversible, because if we run the moving picture backwards the egg will all collect together, and that is obviously ridiculous!" But if we look at the individual atoms themselves, the laws look completely reversible. [...]

I understand that the laws that govern atoms are reversible, but macroscopic things are made of atoms and macroscopic processes do not look reversible at all. How should I interpret this apparent paradox?
Is it the case that most states (combinations of positions and velocities of the atoms) of a physical system lead to an intuitive ending (e.g. egg splattering) and thus intuitive endings are more likely, but if we reverse all the velocities we would arrive at a higly unlikely state?
Or could it be that states that lead to unlikely endings (e.g. egg collecting itselt back together) are so unstable that small sources of randomness are enough to ensure that unlikely endings indeed will not happen? What kinds of randomness must be accounted for? Influences from outside the system? Would it be correct to say that quantum mechanics is a source of randomness in a closed system?
 A: All possible outcomes are equally unlikely. The odds of any exact configuration of particle positions and momentums in a splattered egg are all negligible.
However, some of these configurations result in macroscopic outcomes that look identical, and if you count the number of configurations that appear identical let's call that number the "multiplicity" then they can add together to give you a much higher probability of occurring (although remember each individual state in that multiplicity has a very low probability). So the reason some processes are "irreversible" is because they have a negligible multiplicity compared to other processes.
For example consider an egg was already dropped and splattered on a table. There will be a very large multiplicity associated to the egg just lying there broken (because there are many configurations of random velocities of the particles which just cancel out), but there will be a very low multiplicity assigned to the process of the egg pieces coming together to form an unbroken egg (because this process requires very specific velocities and not many configurations will meet that requirement) so that outcome is negligible.
Maybe if you stare at a broken egg until the universe ends you will eventually see it spontaneously form into a whole egg again, but in the course of a normal human lifespan it is considered to be a statistical impossibility.
So in a sense forward or backward in time aren't special because both are just transitioning between possible configurations of the system according to legal laws of physics.
It only looks special when you let time go in one direction toward higher multiplicity and then reverse the direction of time, because now you've formed a "history" and you're requiring that the system do something which is normally very unlikely, flow toward lower multiplicity. If that history didn't exist then even moving backward in time statistically it would still go toward higher multiplicity which would look perfectly normal.
By the way, you can replace everything I said above about "multiplicity" with "entropy" because they are related by a natural log. Increasing multiplicity is increasing entropy, I just didn't use the word entropy because I think it's less intuitive in this case than talking about the individual particle configurations.
A: 
Is it the case that most states (combinations of positions and velocities of the atoms) of a physical system lead to an intuitive ending (e.g. egg splattering) and thus intuitive endings are more likely, but if we reverse all the velocities we would arrive at a highly unlikely state?

Not intuitive, but consequent, due to the laws of mechanics in this simple picture.
Think of a jig saw  puzzle. It can be randomized very fast. It may even take days to be put together again, just with 1000 bits or so.
One mole of matter contains about $10^{23}$ molecules. These molecules, depending on their interactions between them, which individually are time reversible, have an enormous number of permutations . To get back to a previous state is statistically very improbable. 

Or could it be that states that lead to unlikely endings (e.g. egg collecting itselt back together) are so unstable that small sources of randomness are enough to ensure that unlikely endings indeed will not happen?

No, it is a matter of probabilities, not stability.

What kinds of randomness must be accounted for? Influences from outside the system? Would it be correct to say that quantum mechanics is a source of randomness in a closed system?

Quantum mechanics introduces extra probable states in addition to simple number counting, as it is a theory that predicts probabilities of interactions.
Thus the probabilities of spontaneously getting the same state can be considered zero within the lifetime of the universe.
