From the Silmarillion
Then Sauron failed, and he was utterly vanquished and passed away like a shadow of malice; and the towers of Barad-dur crumbled into ruin, and at the rumour of their fall many lands trembled.
Sauron had bound a lot of his own spirit in the Ring. When "slain" by Isildur, it was only the power of the Ring, anchoring his spirit to Middle Earth that enabled him to return. With its destruction, Sauron's anchor to life was gone. Whether he died utterly, or whether his spirit lived on to take form in a later age I am not certain, but what I am sure about is that he would never again be as powerful.
The nazgul were created by the power of their own Rings, but since the One Ring bound all of the other rings, the loss of the One Ring caused all of the other rings to fail. Galadriel said that the destruction of the One Ring would cause everything that had been achieved by the three elven rings to fail. If this is so, we can expect things created by the Nine rings to fail also. The Nazgul would die.
Virumor has answered the Morgoth question correctly. At the end of the First Age, the other Valar chained Morgoth with a chain created by Aule and a collar forged from Morgoth's own helm, and then flung him into the Void. No doubt they keep an eye on him in there to ensure he does not escape.