Regarding the last question: if you want a polynomial $f(x,y)$ and all its derivatives up to $m$ to vanish at a point $p$, it gives $\binom{m+2}{2}$ independent linear conditions on the coefficients. Indeed, if we assume $p = (0,0)$, then the condition is that monomials of $f$ of degrees up to $m$ vanish, and there are $1 + 2 + \dots + (m+1) = \binom{m+2}{2}$ of these.

Informally, both derived categories and (stable) homotopy types provide a functor from a nonlinear category of spaces to a more linear category. Both formalisms relate to Grothendieck's idea of motives which is supposed to be a universal such functor. Indeed there are conjectural links between motives and derived categories, as well as adjoint functors between triangulated category of motives and A^1-stable homotopy category. Finally, one analogy for Postnikov towers in derived category is the concept of a semiorthognal decomposition.

A simplest example of a normal factorial surface singularity is $E_8$ with equation $x^2 + y^3 + z^5 = 0$. By comments of Pop and Sándor Kovács, there will be no smooth curves on this surface passing through the origin.

Is it a fact that such Fano 3-folds are not smoothable?

In Tschinkel's talk maths.ed.ac.uk/cheltsov/edge2017/pdf/yura.pdf, on slide 15 he says that it is a conjecture of Colliot-The'le`ne that existence of a point plus vanishing of Br(X_K)/Br(K) for all field extensions K/k implies stable rationality for del Pezzo surfaces.

Maybe one thing to add is that Hirzebruch surfaces, that is $\mathbf{P}^1$-bundles over $\mathbf{P}^1$ all have the form $\mathbf{F_n} = \mathbf{P}(\mathcal{O}+\mathcal{O}(n))$, $n \in \mathbf{Z}$, and their diffeomorphism type depends only on $n \mod{2}$: map.mpim-bonn.mpg.de/Hirzebruch_surfaces Finally, $\mathbf{F}_0 = \mathbf{P}^1 \times \mathbf{P}^1$ and $\mathbf{F}_1 = Bl_{x}(\mathbf{P^2})$.

If you also require that $-K_Y$ is ample, so that $Y$ is a Fano threefold, then say in Picard rank one, K3 surface $X$ will be of degree 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, which is part of the classification of Fano threefolds. Beauville generalizes this for arbitrary Picard rank (keeping the Fano assumption): arxiv.org/pdf/math/0211313.pdf

I think the map is not surjective in general. That is, if the image of $\mathbf{P}^1$ lands in the singular locus of $Y$, there is no reason why the map can be lifted to $X$. One can find an example for surfaces, and I think one such example is a surface $x^2 u = y^2 v$ in coordinates $[x,y,z] \in \mathbf{P}^2$, $[u,v] \in \mathbf{P}^1$. Its singular locus is isomorphic to a projective line, and taking normalization yields a connected $2:1$ cover on the exceptional locus, hence there is no lifting for the identity map on the exceptional locus.

Are you asking why Jacobian of a nodal curve is an extension of an abelian variety (in fact, the Jacobian of its normalization) by a torus? This is a result of Oort 1962, link.springer.com/article/10.1007%2FBF01440949, Prop. 2.3.

@ulrich: What exactly do you mean? I think all the `cohomological' data (Artin motives, Galois modules) which can be extracted from $[X]$ will not be able to distinguish fields.

One question I would like to know the answer myself is: what if $\mathbf{L}^n\cdot([\mathrm{K}]-[\mathrm{K'}]) = 0$? Are $K$, $K'$ isomorphic then? Here $\mathbf{L} = [\mathbf{A}^1]$.

...and under not very restrictive conditions on $X$, e.g. quasiprojective we have $K_0(Perf(X)) = K_0(VB(X))$ ($VB$ stands for vector bundles). Hence if an additive identity is checked on vector bundles, it seems to follow for perfect complexes.

One simple-minded comment is that $det$ is a homomorphism from $K_0(Perf(X))$ to $Pic(X)$, hence

Could you please give a reference for $Perf(X)$ being the pullback of all the $Perf(\tilde{X}_k)$?

Thanks! Looking at the proof, it seems that the key nontrivial result inside is that of Anderson, that in this setting vector bundles always split into line bundles: ams.org/journals/tran/1978-240-00/S0002-9947-1978-0485827-5/‌​…, e.g. Prop. 5.1.