Note there are two kinds of movement - electrotonic and salutatory. It is only via salutatory conduction (in myelinated neurons) that it occurs at nodes. In non-myelinated neurons, for all intents and purposes, it occurs along the entire length of the axon.
(1) Yes. It depolarises the nodes ahead, this builds up until they reach the threshold potential.
(2) They are voltage gated channels. Gating refers to the property of being able to open or close them. Voltage is the mechanism that opens or closes them. Once the membrane potential reaches the threshold potential, the action potential is initiated. The threshold potential also marks the point that voltage gated sodium channels are opened, allowing sodium to flood in, depolarising that section of the axon. In other neurons, other ions/channels may be at work but the above is true for your "typical" or "textbook" neuron.
(3) In myelinated neurons, yes. Please attempt the second half for yourself first and i will correct if wrong.
(4) They are regenerated at each node. The depolarisation at one node is still sufficient by the time it gets to the next node to depolarise it and cause a new AP there. This keeps going. Think of a relay race, i run up to you and hand you the baton, you stop for a second and run and pass it along again. It's a little like that.