Why does K+ move out of the cell?
The other answer is a bit misleading.


  "Another cause is the the intracellular K+ concentration"


No, this is exactly the same cause, the differing concentrations is what causes the equilibrium potential. You can think of the equilibrium potential for one ion as "how much voltage does there need to be to prevent this ion from flowing down its concentration gradient." The Nernst equation gives you the equilibrium potential for any one ion. The equilibrium potential is also known as the "reversal potential" because if the voltage exceeds the equilibrium potential, ions will flow the opposite way, against their concentration gradient.


  Don't get confused by the resting potential, because it is only a potential that the cell arranges for itself to maintain a stable environment. This cost a lot of energy and attention in form of protein synthesis to keep this potential. This mechanism is achieved by the Na+-K+ ion-pump


The resting potential is not to "maintain a stable environment" or anything like that. The resting potential is the net potential that a cell reaches due to all of the conductances of various ions and the flowing current of each ion governed by its own equilibrium potential.


  This mechanism is achieved by the Na+-K+ ion-pump, which exchanges those two ions


This is partly true, but again, misleading. The Na+/K+ pump establishes the relative concentration of ions, which only leads to a certain resting potential because the conductances of different ions vary at rest. Because the conductance of K+ is higher at rest than the conductance of Na+, the resting potential is closer to the equilibrium potential for K+. If the Na+/K+ pump was functioning exactly as it does, pumping Na+ out and K+ in, but the membrane was more permeable to Na+, then the resting potential would be positive rather than negative. All that matters for resting potential is relative ion concentrations and conductivity. The Goldman equation is the way to calculate this resting potential.

What is happening to K+ when a cell has a resting potential of -70mV but a K+ equilibrium of -90mV?

Because you need -90mV to 'hold in' the potassium against it's concentration gradient, at -70mV K+ will flow out of the cell. However, by definition, resting potential is the potential at which the net current will be zero. That means that, yes, other ions have to be involved. This is where the Goldman equation is very useful.

Although K+ may be the ion with the highest membrane conductance, there will also always be some leak of Na+ and Cl- ions (usually others ions are ignored because these three are the major players; other ions can be important in some situations). Therefore, if -70mV is rest, there must be at least as many Na+ ions flowing in or Cl- ions flowing out as there are K+ ions flowing out.

You can calculate these currents if you know the conductance of each ion and the equilibrium potential for each ion using Ohm's Law: I = V/R, where R is 1/conductance and V is the difference between the current voltage and the equilibrium potential. Try it out with the Goldman equation! If you use the voltage the Goldman equation gives you (the resting potential) you will find that the net current is zero!

Okay, so far we've answered your questions (1) and (2). Now (3): you state "Otherwise it is not clear why K+ should stop leaving the cell at any moment". It is important to note that equilibrium potential doesn't mean no ions are moving: it means the voltage isn't changing. If you just left the cell at -70mV and waited some time, eventually the concentrations of the different ions would change. This is where the Na+/K+ pump comes in: this ATPase is constantly pumping some K+ out and Na+ in to counteract the leak that occurs at resting potential. Cl- ions move, too, but mostly passively. It's also important to note that very few ions have to move for the potential to change on the order of millivolts. -70mV might sound like a lot, but electrical forces are very powerful, so very few ions relative to all the ions available have to move. 

I don't quite understand what you are asking with (4) but if you can edit the question I will try my best to answer that as well. It's possible I already answered your question by talking about what "equilibrium potential" means for one ion. For a Cl- ion, if the equilibrium potential for Cl- is -65mV, then Cl- will flow into the cell when the membrane voltage is -60mV (tending to make the cell closer to -65mV), and it will flow out of the cell when the membrane voltage is -70mV (still tending to make the cell closer to -65mV).