This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give one possible thread of explanation. The underlying principle is that the appearance of $\rho$ and the "dot" action $w\cdot\lambda=w(\lambda+\rho)-\rho$ in representation theory is closely related to the geometry of the flag variety.
One of the first places one meets $\rho$ (and the dot action) is in the Weyl character formula. A theorem of Kostant shows that the formula can be written as the ratio of two Lie algebra cohomology Euler characteristics. From this perspective, the appearance $w \cdot \lambda$ and $w\cdot0$ in the WCF is ultimately explained by the fact that these are the weights appearing in the weight space decomposition of the relevant Lie algebra cohomology modules, namely $H^*(\mathfrak n, V^\lambda)$ and $H^\ast(\mathfrak n, V^0)$, where $\mathfrak n = \bigoplus_{\alpha>0} \mathfrak g_\alpha$ and $V^\mu$ denotes the irrep of highest weight $\mu$.
We can rephrase this in geometric terms by invoking the "geometric analogue" of Kostant's theorem, i.e. the Borel–Weil–Bott theorem. Kostant's description of the Lie algebra cohomology of $\mathfrak n = \mathfrak g /\mathfrak b^-$ with coefficients in an irrep translates into a representation-theoretic description of the sheaf cohomology of certain line bundles $L_\lambda$ (constructed using integral weights $\lambda$) over the flag variety $G/B^-$ of $\mathfrak g$. Consequently, the dot action shows up in this description, and this time it's accompanied by a shift in degree. This in turn can be explained by Serre duality; the key fact is that canonical bundle of $G/B^-$ turns out to be $L_{-2\rho}$.
So, in some sense, the appearance of $\rho$ and the dot action in the WCF can be thought of as a manifestation of Serre duality.
[N.B. This is a condensed version of my lengthy original answer. The old version can be found in the edit history.]
