Graphs-like data structure with weighted vertices
Your description of an example of your question #2 of having vertex weights and distributing the weight on some $v$ to its neighbours sounds reminiscent of the discharging method famously used in proofs towards the 4-colour theorem. That method, though, had weights assigned to vertices and to the faces of the (planar) graph. But I believe the discharging method has found other applications. Here is a thesis on discharging methods: https://tel.archives-ouvertes.fr/tel-02083632/document
As for your general question #2 of example problems studied with vertex weights, consider looking up

graceful labelings (e.g. https://en.wikipedia.org/wiki/Graceful_labeling)
channel assignment problems (e.g. https://www.sciencedirect.com/science/article/pii/S0166218X04002562)
any vertex selection problem (like max clique, min vertex cover, etc) has a vertex-weighted analogue
problems that choose orderings of vertices (i.e. assign a timestamp to vertices) like in fire-fighter problems (https://www.sciencedirect.com/science/article/pii/S0012365X0600776X)  or graph burning problems (https://arxiv.org/abs/1507.06524)
the definition of threshold graphs involves looking at a vertex-weighted graph and adding an edge between any two vertices whose sum of vertex weights exceeds some threshold value: https://en.wikipedia.org/wiki/Threshold_graph

Here is a paper talking about more application problems of vertex-weighted graphs: https://www.researchgate.net/publication/268065544_Vertex-weighted_graphs_and_their_applications