Dear Mikhail,

I had been hoping someone else would attempt to answer this question, as I have been wondering very similar things lately. (In fact I drove myself crazy for about a month last year trying to work out some solution to what you are asking in the second paragraph.)

I can't answer everything but here is a start:

Write $K = \overline{\mathbb{Q}_{\ell}}$. One already sees the problems with what you are asking for $X_0 = Spec \mathbb{F}_q$. Then the category of constructible $K$-sheaves on $X_0$ is equivalent to the category of finite dimensional $K$-representations of $Gal(\mathbb{F}/\mathbb{F}_q)$, the absolute Galois group of $\mathbb{F}_q$. (The absolute Galois group is generated topologically by Frobenius, and so this is the same as giving a finite dimensional $K$-vector space together with an endomorphism.) [See BBD 5.1.11 for a statement, I think this is explained in Milne, but don't have it at the moment.]

Now, a pure sheaf on $X_0$ is pure of weight $i$ if all the eigenvalues of Frobenius are algebraic integers all of whose complex conjugates over $\mathbb{C}$ have the same absolute value $q^{i/2}$.

Note that here we already see that over $X_0$ a pure sheaf does not need to be semi-simple. Indeed, there is no reason why Frobenius should act semi-simply. (This is one example of what de Cataldo and Migliorini are talking about in Remark 3.1.9 after the Theorem 3.1.8.) I think it is part of the standard conjectures that Frobenius acts semi-simply on the $\ell$-adic cohomology of smooth projective varieties, which as I understand it, is still not known.

I don't know what you mean when you ask:

Does the splitting of part 1 (only!) really requires $\overline{\mathbb{Q}_{\ell}}$-coefficients?

As to your main question, I think that the above example shows that this is too much to hope. Without working over $X_0$ one cannot define what it means to be mixed, and without going to $X_0$ one can't expect the same ext vanishing.

I recently discovered your work on "weight structures" and found it very interesting. I guess you are asking the above, because you would like to argue that one gets a weight structure in the setting of $\overline{\mathbb{Q}_{\ell}}$-sheaves.

There is one setting where I think that one really does get a weight structure. This is in the (at least formally) very similar world of "mixed Hodge modules" on complex varieties. There one has the desired ext vanishing from the outset.