Is bitcoin running over a structured or unstructured overlay? How nodes are being admitted to the system? Does each node have a sequential unique address? 

In other words, are bitcoin nodes considered structured or unstructured?
Bitcoin uses an unstructured network.

The primary reason a network might opt for a structured system is to allow systemic retrieval of information. This is not necessary in Bitcoin, since each node maintains the same information.

Bitcoin does not, however, employ the flooding technique at the start. It uses a set of DNS seeds to get an initial node list, and connects to nodes from that list, before asking them for additional peers. This prevents the bandwidth issues associated with flooding, while maintaining a reasonably decentralized discovery method.

Note that even if all the DNS seeds ceased to exist, new nodes could connect by connecting to any existing node, and then asking for a peer list, and cascading down the peers' peers. In the based, IRC channels were also used for node IP discovery, which again shows how the system can remain functional without the DNS seeds.