**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.