1. His head moves up and down.

2. His head moves down and up.
[#1 is correct. What about #2? do you use 'down and up'?]

3. His head nods.
4. He nods his head.
[Are both the same as #1?]

All are correct. It would be more common for us to say "up and down" instead of "down and up", but both are correct. Three and four are also correct and do mean the same as one and two.