What disadvantages are there when broadcasting a transaction via a 3rd party application? Bitcoin marketplaces could transmit their transactions through e.g. blockchain.info, why aren't they? Do bitcoin marketplaces &amp; selling services run their full node to broadcast a transaction? What alternatives and possibilities are there for broadcasting a transaction, other than through a 3rd party?
The main disadvantages, are you are trusting the third party to actually relay the transaction, that third party can now link you to the bitcoin being moved in your transaction, and you are trusting other people running full nodes to verify your transaction. Your most secure and privacy-centric option would be to use a wallet which relays your transactions through a full node you run yourself. This way you are not trusting a third party to relay your transaction, and running a full node will help verify your own transaction. Some examples of wallets which do this are the wallet built into bitcoin core, Armory, and Electrum allows you to run your own Electrum server, CoPay allows you to run your own server as well, but last time I looked into this it was not a simple process. The drawback to this, is you will need to store the blockchain on your computer, which can require a lot of storage space.

EDIT: The bitcoin wallet mSIGNA, also allows you to use your own full node.