The pools, exchanges and probably other businesses do have full nodes, for their needs and for safety. Others can do this for the sake of decentralization.
Rewards? It can be good and it can be bad. Just look at the masternode coins; I don't like them, I don't find that model as safe and decentralized as Bitcoin.
Sorry if the question seems stupid. But I don't see what need an exchange has to run a full node...