What's the purpose of Karger's algorithm? Suppose I'm given an undirected graph and two nodes: v and u.

If I understand Karger's algorithm correctly, it's used to find a minimum cut of a graph, not the ("the" because there is only one for any given graph) minimum cut such that v is in one part of graph and u is in the other part. 

Why don't we simply find the vertex with the minimum number of edges and remove those edges. Thus, we "partition" the graph into two subgraphs: the node which had the minimum number of edges and the rest of the original graph. What am I missing?
Even in an unweighted graph, your suggested algorithm doesn't work:  take a graph of 6 vertices with triangle a, b, c, and triangle d, e, f, with an edge between c and d.  Vertices a, b, e, and f each have minimal degrees of 2, but cutting the graph between the two triangles gives a cut of 1.