I'm also unsure of how blockchain.info "wallets" actually work.
It's connected to full nodes and pushes transactions through them (in simple).
Can you please explain a bit more? Aren't the full nodes belong to them?
It was also suggested that transactions might be not validated? How can a computer be connected to the block chain (node) without fully enforce all of the rules of Bitcoin and NOT using a full node?
Jokes aside, Blockchain.info probably have full node that syncs blockchain somewhere that performs all of the usual functions of full node.
However they also have software that parses information from actual blockchain, writing it into searchable database (SQL or something similar). Your API call to blockchain.info most likely does not interact with actual node in any way, it only operates with information stored in their database.
I have no idea how their on-line wallet operates, but chances are in very similar way. They only use their node to push your transactions to the web. All tx generation and signing happens in some sort of backend app.