Computation (example) with vector fields in $\mathbb{R}^3$.

Question

I'm trying to apply some results about metric measure spaces for my thesis but I'm not understanding how to go further.

I have $\mathbb{R}^3$ with the two vector fields $X$ and $Y$ that define the Heisenberg group:\begin{align*}X&:=\partial_x+\frac{y}{2}\partial_z,\\Y&:=\partial_y-\frac{x}{2}\partial_z.\end{align*}I'm considering now the distribution $\Delta:=\mathrm{span}(X,Y)$ and I put on $\mathbb{R}^3$ the Riemannian metric $g$ for which $X$, $Y$ and $[X,Y]=\partial_z$ are orthogonal. If I consider the submanifold $\{x=0\}$ given so in implicit form:

1. the gradient of $f(x,y,z)=x$ (that defines implicitly the submanifold) how is defined? I don't have $\partial_x,\partial_y,\partial_z$;
2. the gradient of $f(x,y,z)=x$ as a vector field belongs to $\Delta$? How can I express it as a combination of the base vector fields?
3. the "horizontal gradient" is defined as $\nabla_Hf:=(Xf,Yf)$. So how can I use it (it has 2 components not 3, or I'm wrong about something?)?