homework - Problems in the concept of ionic equilibrium

While trying to understand the solution of a problem given in my text book, I realized I'm having some difficulty with the solution. The problem is as folows:

The ionization constant of HF is $3.2 \times 10^{-4}$. Calculate the degree of dissociation of HF in its 0.02 solution. Calculate the concentration of all species present $\ce{H3O+ , F+}$ and $\ce{HF}$ in the solution and its PF.

In the solution of this problem, the equation is given as

$$\ce{HF + H2O <-> H3O+ + F-}$$

The concentration of at the time of equilibrium are given as:

$$[\ce{HF}] = 0.02 - 0.02x, \: \ce{[H3O+]} = 0.02x, \:\ce{[F- ]}= 0.02x$$

I have the following doubts:

1. Why we are not adding the contribution of water to the $\ce{H3O+}$ ions?


2. Why we are assuming that the value $0.02x$ is dissociated from $\ce{HF}$ and not just $x$.?