For 360s, the definitive way to do it is hold the "walk" button, then do a half circle with the right analog, starting at the side you're running towards. So, right, bottom right, down, bottom left, left ---> spin.
For free kicks, don't let the initial bar go all the way to the end, it will go too high. Try to stop it first in the middle, then stop it in the green for accuracy.
For headers from goal kicks, you have to press the shoot button. You can even do them with no one around you (hello Fifa 2003!).
Still working on crosses. Noticed you can charge up shots for players waiting in the box for an incoming cross. Gotta experiment more...