If your Weil cohomology theory is étale cohomology, then (2) is easy. Indeed, we have specialisation isomorphisms (for $\ell \neq \operatorname{char} \kappa(s)$) [SGA4$_3$, Exp. XVI, Cor. 2.2]
$$H^i_{\text{ét}}(\mathcal X_{s'},\mathbb Q_\ell) \stackrel\sim\to H^i_{\text{ét}}(\mathcal X_s, \mathbb Q_\ell)$$
as well as specialisation maps [Ful98, Ex. 20.3.5]
$$\operatorname{CH}^i(\mathcal X_{s'}) \to \operatorname{CH}^i(\mathcal X_s)$$
that are compatible with the cycle class maps $\operatorname{CH}^i \to H^{2i}$. Thus, if you have a cycle on $\mathcal X_{s'}\times \mathcal X_{s'}$ inducing the Lefschetz operator on $\mathcal X_{s'}$, then its specialisation induces the Lefschetz operator on $\mathcal X_s$. The same argument works whenever the Weil cohomology theory has some sort of specialisation isomorphism.

There is no hope to prove (1), because the specialisation map on Chow groups is usually not surjective. For example, the Künneth standard conjecture (Conjecture C) is known over finite fields and algebraic extensions thereof [KM74, Thm. 2(1)], but not over any other type of field.

**References.**

[Ful98] Fulton, William, *Intersection theory*. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge **2**. Berlin: Springer (1998). ZBL0885.14002.

[KM74] Katz, Nicholas M.; Messing, William, *Some consequences of the Riemann hypothesis for varieties over finite fields*, Invent. Math. 23, 73-77 (1974). ZBL0275.14011.

[SGA4$_3$] Artin, M.; Grothendieck, A.; Verdier, J.L.; et al, *Seminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas (SGA 4). Vol. 3: Exp. IX--XIX*, Lecture Notes in Mathematics **305**. Berlin-Heidelberg-New York: Springer-Verlag (1973). ZBL0245.00002.