1) No. You can always mirror-reflect your definition across the plane that defines over/under.
2) I think so. Take a look in the Chmutov, Duzhin, Mostovoy survey on the arXiv. On page 92 they have the first ten non-trivial Vassiliev invariants, computed on sufficiently-many knots. If what you're looking for isn't near there, they likely have a reference for it.
3) Type one Vassiliev invariants are trivial for knots, and for links I believe linking number is the only non-trivial one. I'm not following your argument but I'm also not seeing where it's going. Perhaps this is what you're trying to prove?
