Motivation to run full nodes
I'm not sure I understand your question, but: No, Bitcoin nodes do not announce their own Bitcoin addresses as part of the network protocol.  

One could imagine modifying the protocol so that they did, but it would raise anonymity concerns: currently, by design, it is difficult to match an address with its owner's IP address.  If you knew at least one address that belonged to a particular node, you  might be able to look at other transactions in the block chain and deduce other addresses belonging to that node.  And ISPs usually have records that determine the identity of a user from their IP address.