Jones polynomial in Bar-Natan’s paper “On Khovanov’s Categorification of the Jones polynomial”

Question

I’m reading Bar-Natan’s paper “on Khovanov’s categorification of the Jones polynomial”, I had previously been reading Lickorish’ book to have a good understanding on the Jones polynomial before diving into homology; however Bar-Natan’s construction in the very beginning of the paper (section 2) differs substantially from that I had seen basically everywhere else, I will state my issues and suspicions now:

1. Lickorish defines the Kauffman bracket polynomial by the three rules: whereas Bar-Natan states that the rules are: $\langle \emptyset\rangle=1$ (which together with the next one is equivallent to the first rule of Lickorish’) and the following two:So my first question is why would these two be equivalent (or at least almost equivalent, up to some multiple of $q$ or a change of variable, I’ll explain later), even if I suppose that $-A^{-2}=q$ (which would help make sense of my next question about the trefoil) the third rule doesn’t add up to me. Also why would we prefer this version of the skein relations over the more widely used and known?
2. Again guided by Lickorish and almost everyone else, once we have the bracket, the Jones Polynomial is just a couple of changes away: according to him, $J(L)=\left((-A)^{-3w(D)}\langle D\rangle\right)_{t^{1/2}=A^{-2}}$ ($D$ is just a diagram for $L$), but for Bar-Natan, $J(L)=\displaystyle\frac{(-1)^{n\_}q^{n_{+}-2n\_}}{q+q^{-1}}\langle L\rangle$, which again is not clear to me why it’s almost equivalent, and I say almost because he immediately illustrates the calculation of the Jones polynomial for the left trefoil which he finds to be equal to $q^2+q^6-q^8$, but everywhere else one finds $J(\:\overline{3_1}\:)=t+t^3-t^4$ which is the same but with every power divided by two. So why does Bar-Natan’s version not equal the one universally accepted.
3. And lastly, I tried to find the original normalization factor hidden in the paper’s version, on that note I imagine when he divides by $(q+q^{-1})$ is because every disjoint union of $k$ cicles has the bracket of the unknot raised to $k-1$, and earlier on he describes that for every complete smoothing of a knot one takes the bracket of the unknot raised to the amount of components resulting, so we have to take one out of each power, but in that case, why not just assign to every smoothing the bracket of the unknot already raised to the right power? Also, I tried to find the writhe of the diagram hidden in his notation of $n_+-2n\_$ but again I can’t make sense of it.

I’m also aware of the state sum model for the bracket so if anyone can give me some intuition on this questions based on that version of the bracket I will be happy.