Fast deletion / contraction in combinatorial embedding
This question is incomplete without specifying what information about the graph as it changes you want your dynamic graph data structure to output or support queries for. But the following paper is likely relevant, even though it is described in a more general setting of combinatorial embeddings in arbitrary genus rather than just planar. It definitely supports both contractions and deletions, as well as their reverse operations, in logarithmic time per operation.

Dynamic generators of topologically embedded graphs.
D. Eppstein.
arXiv:cs.DS/0207082.
SODA 2003, pp. 599-608.

As for the second question: I don't see how to handle multiple adjacencies in general, but it's easy to get rid of bigons (multiple edges with nothing between them) as they can only come from the two faces that are on either side of a contracted edge or from the face that surrounds a deleted edge. That should be sufficient for many purposes since getting rid of the bigons ensures that the remaining graph has a number of edges proportional to its number of vertices.