Is the Cayley graph of a free group dense on the Poincare disc?

Question

In other words, every point inside the disc corresponds to a word (possibly of infinite length) of the free group; Is that correct?

With this embedding:

Answer 1

Not true. The free group on two (or finitely many) generators has countably many elements, the hyperbolic disc has uncountably many.

If you consider also the edges of the Cayley graph it is still not true. Look at the circle around the center of the hyperbolic disc which intersects the Cayley graph in the four points $a,b,a^{-1},b^{-1}$. The interior of this circle is not reached again, as all edges point outward. Hence the only points from the Cayley graph contained in this circle lie on the four main edges adjacent to the center (neutral element of the free group) that you already see in your drawing.