Is the quantile function of a differentiable function of absolutely continuous random variables differentiable?

Question

Let assume $n$ absolutely continuous random variables $X_1, \dots X_n$, and a differentiable function $$g: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}; (x_1,\dots,x_n \, ; \, y) \mapsto g(x_1,\dots,x_n \, ; \, y).$$

I am wondering if:

1. $\forall y \in \mathbb{R}^m$, is the random variable $g(X_1,\dots,X_n \, ; \, y)$ absolutely continuous?
2. $\forall y \in \mathbb{R}^m$, is the cdf of $g(X_1,\dots,X_n \, ; \, y)$ denoted by $F(\cdot \, ; \, y)$ continuous and strictly increasing on its support?
3. If for a given $\alpha$, $Q_\alpha$ denotes the quantile function at the point $y \in \mathbb{R}^m$, i.e. $$Q_\alpha: \mathbb{R}^m \to \mathbb{R}; y \mapsto F^{-1}_{\alpha} (y) = inf \left\{ z \in \mathbb{R} \, | \, \alpha \le F(z \, ; \, y) \right\}.$$ Is the function $Q_\alpha$ differentiable in $y$?

I believe that if 1. holds then so does 2. But I have no clue if 3 would hold too.

Thank you in advance for your help!