Why is N₂ stable but HCN and C₂H₂ unstable?
I think this question can be answered reasonably well with thermodynamic considerations.  When we say something is stable, it is (or should be) assumed that the compound does not engage in a chemical reaction.  It is important when we discuss the stability of compounds to focus not only on the compound itself but what the products of the reaction are.

The three triple bonds you mention ($\ce{N#N}$, $\ce{C#N}$ and $\ce{C#C}$) all have very high bond dissociation energies (from various sources such as this one they are $\approx950, 940\text{ and }960\:\mathrm{kJ/mol}$) so all of these triple bonds require a large amount of energy to break.  

Now, when we think about the stability of acetylene, I suspect most people are thinking about its stability towards ignition (which has been studied, by the way).  That gives us a reaction we can explore, namely the combustion of acetylene in the presence of oxygen to produce carbon dioxide and water.  WolframAlpha can calculate the thermodynamics of this reaction, giving a result of $-2600\:\mathrm{kJ/mol}$; clearly a very exothermic reaction.

Now let's look at the combustion of nitrogen in the presence of oxygen, which could form nitrogen dioxide.  Again, WolframAlpha (Note, if you happen to play around with WA, make sure you put your reaction in quotation marks or it won't be interpreted properly) gives us a result of +66 kJ/mol or an endothermic reaction.  

What do we make of these results?  Focusing on the products we see that for the acetylene combustion, there is a large amount of energy released when creating the products.  The same is not true for the products of nitrogen combustion.

In summary, when considering stability, we must think about we mean by the word, since that meaning will lead to a chemical reaction, and the tools we use to describe chemical reactions help us quantify that stability.