One concrete reason: Spherical functors are only defined at the level of DG categories: see arxiv:1309.5035. This is basically because functorial cones exists only in the dg setting.

As $\operatorname{gcd}(2^p-1,2^q-1)=1$ if $p,q$ are distinct primes, it's at least the product of $2^p-1$ where $p$ ranges over the primes at most $n.$ I think this should give you an asymptotic lower bound of $2^{cn^2/\operatorname{log}(n)}$ where $c$ is any constant less than $\frac{1}{2}.$

The compactified total space of the chosen bundle on $\mathbb{P}^1\times\mathbb{P}^1.$

I know little about Gromov-Witten theory, but why should we expect the double covers to contribute 0 (as opposed to having to do some sort of excess intersection formula)?

Take a bundle on $\mathbb{P}^1\times\mathbb{P}^1$ so that the restriction to ${x}\times\mathbb{P}^1$ depends on $x$. Now take the total space of this bundle and compactify it. Then taking $C_i=x_i\times\mathbb{P}^1,$ I believe this construction gives for any pair of vector bundles $M_1,M_2$ on $\mathbb{P}^1$ with the same degree two curves $C_1$,$C_2$ with $N_i=\mathcal{O}\oplus M_i$. Now I believe this immediately gives a counterexample to your statement by taking $M_1=\mathcal{O}(-1)\oplus\mathcal{O}(1)$ and $M_2=\mathcal{O}\oplus\mathcal{O}$, but my knowledge of nef vector bundles is bad...

I'm not an expert on cubic fourfolds, but it seems to me like you could maybe unwind the proofs in Hassett's paper "Some Rational Cubic Fourfolds" using the explicit descriptions of surfaces contained in cubic fourfolds (e.g. in the examples section of his "Special Cubic Fourfolds") to get an answer to this?

Not for $n=2$ and $i=1$; take $X$ the variety of lines on a smooth cubic threefold, then this is basically the main result of the Clemens-Griffiths proof of irrationality of smooth cubic threefolds. My guess is that for $i$ at least $2$ this is a hard question.

The answer to mathoverflow.net/questions/7793/… gives a universal property for Z (namely, that of being a minimal generator)

@QiaochuYuan: I just checked and you're right... I'm suddenly really confused as to how I've gotten away with thinking that it was left module for a while...

@QiaochuYuan: I'm admittedly mostly familiar with the classical case where $\operatorname{Ext}^i(k,k)$ vanishes, and there $RHom(k,-)$ gives you a left module; why is this case different?

@JimHumphreys: I am confused; Soergel, in his remark 4.4, calls the polynomials p_{B,A} "periodic polynomials" (and refers to the generic decompositions pattern paper). Has the notation changed since Soergel wrote that paper? I am not too familiar with the field.

Thanks! I have made the changes you suggest. I'm working here with affine Weyl groups, though having an answer for affine type A would already be really good.

Correction: It's a bit more complicated than that (the quintic threefold will be singular, and this 3-fold will be a resolution (see arxiv.org/abs/math/0508127v2). In any case it looks like the BPS numbers of this Calabi-Yau threefold have been calculated in arxiv.org/abs/1101.2746v2. In particular, I believe that 4-1.1 shows that there are 100 lines on a generic such threefold (and in particular that there are at least 100 on any such threefold).