In LPA*, how are predecessors/successors of a vertex defined?
If you have an undirected graph, you can convert it to a directed graph by including edges in both directions: for each undirected edge $(u,v)$, add directed edges $u \to v$ and $v \to u$.  So, if you have undirected edges, imagine applying this transformation and then running LPA* on the resulting graph.  That should make it clear what to do.

Alternatively: in an undirected graph, Pred(u) is the set of all edges incident on $u$, and so is Succ(u) (there is no distinction between them in an undirected graph).  So, yes, they can overlap.

The cost of the edges are irrelevant.