Keeping the definition of norms in mind, given where a norm is defined (like the Euclidean space, or on a set of functions (like mappings from $C^{\infty} \to C^{\infty}$), or any other valid space) unique?
In other words, for instance, are the norms $||K|| = \sup_{I \subset R^n}(f - g) ; \ \ f,g:R^n \to R^m;\ \ f,g \in C^{\infty} $ or $||K||=\sqrt{x^2 + y^2}, \ \ x,y \in R \ $ unique where they are applied? Is there any other norm that passes the three properties of norms and yet different from ones I've mentioned?
First of all, you need to make precise what it means for a norm to be unique; or rather when two norms are equal.
On an infinite dimensional vector space, there exist norms which are not equivalent (in the sense described here). Examples of such norms are the $p$-norms. For a function $f: \mathbb{R} \rightarrow \mathbb{R}$ s.t. the following integral exists, the $p$-norm with $1 \leq p < \infty$ is defined by $(\int \vert f(x) \vert^p dx )^{1/p}$. For any two different $p$ these norms are not equivalent.
However, on a finite dimensional (real) vector space, all norms are equivalent.