Sigh

So therein lines the key... Just because something is properly ordered, doesn't imply it is valid. And once you add transaction validation, the level of performance will inevitably drop significantly.

The order is enough because you know who paid which token first.

Every transaction explicitly includes its new output state. This state includes everything that must be modified in an atomic manner. If the UTXO were an exchange order book, then the state would be the entire order book. What works for Bitcoin currency (a short script and a balance), doesn’t work for anything that is even slightly more complex or that refers to more data.

Yes. "the state is the entire order book". The state is determined by effectlivy sending everyone the whole hashgraph. This is called virtual voting because you already know how someone would vote if you see the state of the world.

From your own link:

HashGraph uses two special techniques to achieve fast, fair and secure consensus.

Gossip about Gossip
Virtual Voting
Gossip about Gossip basically means attaching a small additional amount of information to this Gossip, which are two hashes containing the last two people talked to. Using this information, a Hashgraph can be built and regularly updated when more information is gossiped, on each node.

Once the Hashgraph is ready, it is easy to know what a node would vote, since we are aware of information that each node has and when they knew it. This data can thus be used as an input to the voting algorithm and to find which transactions have reached consensus quickly.

http://www.thewindowsclub.com/what-is-hashgraph