I am working in Mathematical Finance and know that the Black-Scholes PDE is degenerate at $x=0$ (I assumed that this was because at 0 the convection and diffusion terms vanish and one is left $V_{t} = rV$).
Is there a more precise way of defining this ?
Secondly, is there a reason that one needs weighted spaces to consider finite element methods other than the fact that the weights arise naturally when deriving the weak formulation?
Consider a second-order differential operator $$\mathcal{L}u\equiv \sum_{i,j=1}^n a_{ij} u_{ij}+\sum_{i=1}^n b_i u_i + cu$$ where I am using subscripts on $u$ to denote derivatives. In your case, $$\mathcal{L}V\equiv rSV_{S}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}-rV.$$ $\mathcal{L}$ is said to be elliptic at a point $x$ if for every nonzero real vector $\xi$, $$\sum_{i,j=1}^n a_{ij}(x) u_{ij}(x)\xi_i\xi_j>0.$$ It is degenerate elliptic at a point $x$ if the above inequality holds with equality.
Lastly, it is worthwhile to mention that uniform ellipticity is defined by changing the inequality to $>C$ for some $C>0$ for all points in the domain of interest. Note that the Black-Scholes operator is uniformly elliptic on any $[0,\infty)\setminus B_r(0)$ where $r>0$.
(I do not know the answer to your second question)