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Artin gluing works without change but the ugly sheaf of rings $i^{-1} \mathcal{O}_X$ appears. The structure sheaf of the formal completion can be understood as an approximation to this sheaf (which however does the job only for coherent sheaves). According to Fujiwara-Kato, one can build an analog with henselizations or Zariski localization taking the role of formal completions, but I haven't worked with those.
P.S. This works only for quasi-coherent sheaves. For example, the module $\mathbb{Q}_p/\mathbb{Z}_p$ becomes zero after tensoring by $\mathbb{Q}_{p}$ and has zero completion.
A good analog is to consider the formal completion (so $\mathbb{Z}_p$ instead of $\mathbb{F}_p$ in your example). See Artin "Algebraization of Formal Moduli: II. Existence of Modifications", Theorem 2.6 for the noetherian affine case, Beauville–Laszlo for the non-affine case with vector bundles, and Ben Bassat–Temkin for a global case.
Fiber length should be lower semicontinuous for flat quasi-finite maps. To see this, you can reduce to the case of base being a dvr, and apply Zariski's main theorem to compactify the map to a finite flat map.
Yes (with a single blowup) if the log resolution is projective (and $X$ is quasi-projective), by II 7.17 in Hartshorne. No in general, for e.g. Hironaka's example (Appendix in Hartshorne) is a proper birational map to $\mathbf{P}^3$ with smooth non-projective source, which can't be a blowup or a sequence of such.
No! Consider the point $1/p$ on the affine line. Yes if $X$ is proper, by the valuative criterion. In a way, $p$-adic completion removes all points for which the valuative criterion fails.
Perhaps you already know this: By Bittner's theorem, $K_0({\rm Var}_K$ is isomorphic to the ring generated by smooth projective varieties modulo relations $[X] - [Z] = [{\rm Bl}_Z(X)] - [E]$ where $Z\subseteq X$ is a smooth closed subvariety and $E\subseteq {\rm Bl}_Z(X)$ is its preimage in the blowup. It follows that every motivic measure on smooth projective varieties (satisfying the blowup relation) extends uniquely to a motivic measure on all varieties, satisfying the scissor relations. So in particular the Hodge polynomial extends uniquely, giving the virtual Hodge polynomial.
I think for perfect complexes we get the pull-back category. For coherent sheaves things might be more complicated since the pull-back functors might not even be defined on the bounded category (because of unbounded Tor).
What I'm saying is that this enriched variant takes values in the category ${\rm FGL}_+(\mathbf{Q}) := \varinjlim_N {\rm FGL}(\mathbf{Z}[1/N])$ rather than ${\rm FGL}(\mathbf{Q})$. The natural functor ${\rm FGL}_+(\mathbf{Q})\to {\rm FGL}(\mathbf{Q})$ is not an equivalence, precisely because the logarithm/exponential series have too many primes in denominators. (A similar construction should work over any field of char. 0, with colimit taken over all f.g. $\mathbf{Z}$-subalgebras.)
(cont.) ...of $E$ (point counts over all finite fields with large enough characteristic) and hence (by Chebotarev density and Tate conjecture) should be able to determine the curve $E$ up to isogeny.
It is true that the formal group over a field of characteristic zero is a very lossy invariant. However, things are maybe not so bad over a number field (say $K=\mathbf{Q}$). Spread out your elliptic curve to a group scheme over $\mathbf{Z}[1/N]$ for some $N$ and take the associated formal group over $\mathbf{Z}[1/N]$. This should actually be an invariant of $E$, the "formal group up to change of coordinates which has finitely many primes in denominators". And as far as I know (somebody here will be able to give references) by work of Honda (?) this formal group remembers the $L$-series ...
