It uses it's own crypto. It's not really a crypto -currency- in the sense that it's meant to be used for payment or to store value. It's more of a crypto -token- that's meant to provide some limited utility in it's ecosystem. Like an arcade token in an arcade, you can use it to play the games but that's about it. Likewise the session token can be used to get some extra functionality within the network, like registering custom names on it's dns like service that can be used to add new contacts instead of the long default user hash or as a stake if you want to run a node. The functionality is fairly limited right now but the devs plan to expand it soon. People also sometimes use these kind of tokens as a stock of sorts, so if the service/network becomes popular the value of it's "stock" can grow so it can be used as an investment (personally I wouldn't recommend that but whatever floats your boat [not a financial advice btw]). The node operators profit from selling these tokens to whomever wants to buy them.

No, DHT is just a way of determining paths and priority of value lookup by key in the network, so that the load were distributed predictably, while allowing you to find, well, what you are looking for. BTW, while everybody uses Kademlia with modifications, I'd argue that Chord is better for anything related to security and anonymity.

Storage and serving of anything big is another thing. I take it you mean that I2P nodes cache messages relayed via them when the target node is unavailable. That doesn't have anything to do with DHT.