Dividends can be distributed just like dividends are distributed in current companies - on a pre-declared date, everyone holding the 'stock' will get the dividend.
Crypto-currencies are really backed by the value of services that they provide.
USD is a collectible too, without any "backing". Some time ago USD was gold backed, meaning it was a promise to give you a certain amount of metal from the bank. Since 1972 (actually, for average Joe it all started even earlier, at Bretton Woods agreement) USD is officially not backed by gold. It's a pure collectible protected from forgery by law enforcement (both paper bills and banking transactions) and also protected from competing collectibles by various regulations and controls. As long as govt does not print dollars too quickly, it remains being a scarce asset which can be used as a collectible. Enforced requirement to pay taxes only works as long as dollars are demanded on the open market as a scarce collectible. If dollars are only needed as tokens to be purchased from govt with real goods (or bitcoins) to be paid back to it, then they have no meaning as money and it's going to be just a fancy way to pay taxes in something else.
[1]: "But neither does the US Dollar!" Yes, it does; the US government acccepts taxes only in the form of dollars, and has the authority to force many other entities to render unto it taxes. Upon this basis is the value of government fiat currencies built, and it's why currencies follow the fate of their issuing government quite precisely. (The "it has no backing" model would predict that the currency could float on regardless, which they observationally do not do.)