Dear Kestutis,

your question is *integral* in two ways: first of all, you would like to consider schemes over a whole DVR instead of the generic fiber only. Secondly, you would like to have a comparison theorem between two $\mathcal{O}_\overline{K}$-modules and not only between $K$-vector spaces. The two are tightly connected since $H_\text{dR}(\mathcal{X})\otimes K=H_\text{dR}(X)$, if $X$ denotes the generic fiber of $\mathcal{X}$.

As for the first question we have the so-called Crystalline Conjecture proven by Faltings and many others. It says that if $\mathcal{X}$ is proper and smooth over $\mathcal{O}_K$ (I keep your notations) with generic fiber $X$ and special fiber $\overline{\mathcal{X}}$, then
$$
B_\text{cris}\otimes_{K_0}H_\text{cris}^m(\overline{\mathcal{X}})\cong B_\text{cris}\otimes_{\mathbb{Q}_p}H_\text{et}(X_\overline{K},\mathbb{Q}_p)
$$
where $B_\text{cris}$ is Fontaine's ring of periods. If you extend scalars to $B_\text{dR}\supset B_\text{cris}$ you get back the comparison isomorphism that you quote because $B_\text{dR}$ admits a filtration whose associated graded ring is $\bigoplus_{q\in\mathbb{Z}}\mathbb{C}_K(q)$; and because of a basic result by Berthelot and Berthelot-Ogus saying that crystalline cohomology (which is the group $H_\text{cris}(\overline{\mathcal{X}})$ in my equation above and is a cohomology theory of the special fiber with coefficients in $W(k)$, the Witt vectors of the residue field $k$ of $\mathcal{O}_K$) is isomorphic to de Rham cohomology *after you tensor both with $\text{Frac}(W(k))$*. You could give a look at

1) J.-M. Fontaine, *Représentations $p$-adiques semi-stables*, in *Périodes $p$-adiques*, Astérisque 223, Section 6.1 for a discussion about the Crystalline Conjecture

2) P. Berthelot and A. Ogus, *Notes on crystalline cohomology*, Princeton University Press, Chapter 7 (the very last theorem) for the isomorphism $H_\text{cris}\cong H_\text{dR}$

Let me also remark that much of the above can be generalized assuming only that $\mathcal{X}$ be semistable instead of smooth (by Tsuji).

The second question is more subtle. As I mentioned, already the isomorphism $H_\text{cris}(\overline{\mathcal{X}})=H_\text{dR}(\mathcal{X})$ is false in general: it is true if $K/\mathbb{Q}_p$ (or $\mathcal{O}_K/\mathbb{Z}_p$, equivalently) has absolute ramification index $e\leq p-1$. In this case, at least, we dispose of an interpretation of the de Rham cohomology of $\mathcal{X}$ in terms of crystalline cohomology. We also have an "integral structure" $A_\text{cris}\subseteq B_\text{cris}$ and one can hope to have something like
$$
A_\text{cris}\otimes_{W(k)}H_\text{cris}^m(\overline{\mathcal{X}})\stackrel{?}{\cong} A_\text{cris}\otimes_{\mathbb{Z}_p}Hˆm_\text{et}(\mathcal{X}_{\mathcal{O}_\overline{K}},\mathbb{Z}_p)
$$
The problem is that $A_\text{cris}$ (and even $A_\text{cris}[1/p]$!) is much smaller than $B_\text{cris}$: roughly speaking, it cannot detect "negative Hodge-Tate weights".

[EDIT] As Keerthi Madapusi Pera and Matthew Emerton commented, in some special cases the above isomorphism holds true. They give precise references in the comments.