Why do we want norms to follow triangle inequality What purpose does Triangle inequality serve while defining norms? 
Is it important to render it as a distance metric?   
What would happen if we let the condition loose?
 A: If one defines a map $\|{\cdot} \| : V\to \Bbb{R}$ that satisfies all of the conditions of being a norm, but we are less strict with the traditional triangle inequality; for example
$$\|x+y\|\leq K(\|x\|+\|y\|) $$
for some fixed $K>0,$ then we call this a quasinorm. There is much to consider of such a definition, see: http://www.ams.org/journals/bull/1945-51-01/S0002-9904-1945-08273-1/S0002-9904-1945-08273-1.pdf
When $K$ is not fixed there isn't clear answer as to what your function is actually telling you about $V$. What is desirable about the triangle inequality is that it furnishes us with a metric, that is, if you define $d(x,y)=\|x-y\|,$ then you establish $(V,d)$ as a metric space as you have already pointed out.
Others have pointed out various other conditions, all of which are contingent on the topology endowed by the metric defined by the norm. So what happens in the end when you define a map which ignores the triangle inequality, you ignore the metric topology, and in particular, you ignore any geometric aspects of your vector space which may be of use. Moreover, metric spaces can greatly reduce the difficulty of many arguments, and may even allow for analysis (depending on the field $V$ is defined over).
A: Properties whose proofs rely on the triangle inequality include:


*

*A ball is convex.

*Vector addition is continuous.
