Most of the definitions on flux and flux density, show a plot consisting of a positive charge emanating a field, and describe that as the number of field lines decrease, the field strength decreases. My question is, If an electric field (which is a vector field) is defined at every point in space, how does the density fall as we travel away, as there is a vector associated with every point in our "unit area" where ever it's considered. So how does the density "fall"?
Field lines are a vizualization tool. Each point on the line represents the direction of the field at that point. The lines are usually drawn so that their density corresponds to the field amplitude.
However, field lines are not particularly useful for calculations. Yes, a vector field is defined with a direction and amplitude at each point in space. The field amplitude falls off with distance from its sources. For a point charge, the field amplitude falls off as $1/r^2$. Fields from more complicated sources are found by superposition.
Electric flux is not generally measured in lines, but is caclulated by integrating over a surface the field amplitudes crossing perpendicular to the surface. Dividing by the area gives average flux per unit area, and taking the limit as the area goes to zero determines the local flux density.
Edit: Added to address comment regarding Volume II, Chapter 4 of the Feynman Lectures.
The number of electric field lines drawn from a charge ($q/\epsilon_0$) represents field strength. As Feynman says, '... strength of the electric field will be represented by the “density” of the lines.' You can count lines, but to quantify flux, you must know the field strength their density represents.
Feynman ends the chapter with the following quote.
"The field-line picture has its uses, however, so we might still like to draw the picture for a pair of equal (and opposite) charges. If we calculate the fields from Eq. (4.13) and the potentials from (4.24), we can draw the field lines and equipotentials. Figure 4-13 shows the result. But we first had to solve the problem mathematically!"