Question 1. The answer is yes, and the classical example is as follows. Is it possible to find a one-parameter family $\psi \colon \mathcal{X} \to \Delta$ such that $X_0$ is isomorphic to the cone over a rational normal curve $C_4 \subset \mathbb{P}^4$, whereas $X_t (t \neq 0)$ is isomorphic to a smooth rational normal scroll $X \subset \mathbb{P}^5$.
The surface $X_0$ is $\mathbb{Q}$-factorial, because it has only a cyclic quotient singularity of type $1/4(1, \, 1)$. However, your third condition is not satisfied since for all $t \neq 0$ we have $$9=K_{X_0}^2 \neq K_{X_t}^2=8,$$ whereas (as an immediate consequence of Condition 3) any $\mathbb{Q}$-Gorenstein smoothing preserves the self-intersection of the canonical class.
Question 2. The answer is still yes. If $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, then there exists an integer multiple $mK_{\mathcal{X}}$ of it which is a Cartier divisor. But therefore the divisor $$mK_{\mathcal{X}/ \Delta}= mK_{\mathcal{X}} -m \psi^* K_{\Delta}$$ is also Cartier.
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${}$ References.
- The deformations of cones over rational normal curves are described for instance in
J. Stevens, Deformations of Singularities, Lecture Notes in Math. 1811, Springer 2003.
- For the theory of $\mathbb{Q}$-Gorenstein smoothing of surface singularities, see
M. Manetti, Smoothing of singularities and deformation types of surfaces, in Symplectic 4-manifolds and algebraic surfaces, 169–230, Lecture Notes in Math. 1938, Springer 2008
and the references given therein.