Does a bitcoin node broadcast that it is full synced? When a bitcoin core node is fully synced, does it broadcast to the rest of the network that is has finished syncing? If yes, where in the bitcoin-core source code is this implemented?
No. The nodes notify only about block height in the beginning of connection in version message. When node wants to notify about new block/tx it received/created, it uses inv message.

There is no point to notify other peers about downloading full blockchain. Other nodes have nothing to do with such information.