How is a transaction broadcasted?
it is more like 3.
The client would send the tx to a full node (let aside the complexities of SPV clients), and this full node would verify the tx by some pre defined rules. If ok, this full node transmits to other full nodes. 
The transaction only gets valid, if it was mined into a block. So the server you mention can be several blocks behind, but still can transmit the tx to other nodes. A very good overview of the way this happens is here: 


  http://chimera.labs.oreilly.com/books/1234000001802/ch06.html


also the chapter on transactions will help to clarify the picture.