1) As others have said, $g(x,y,z) = \nabla f \cdot (1,1,1)= (f_x, f_y, f_z)\cdot(1,1,1)$ is a way to write your function. I don't think there is a better way to write it than what you have already, as it's straight to the point.
2) & 3) It depends. If you know each of $f_x, f_y,$ and $f_z,$ then you can do it up to a constant by integrating, say, $f_x$ first with respect to $x,$ then $f_y,$ and so forth, comparing the unknown functions that will arise. This is akin to finding a function $f$ such a given vector field ${\bf F} = \nabla f$. On the other hand, if you don't know the functions in the first place, consider the case where $f = 2x - y - z,$ where $f_x + f_y + f_z = 0.$ There are a great many functions whose partial derivatives sum to zero, so how would you find $f?$ No matter what the derivatives sum to, $f$ can't be found with surety.
4) Not that I know of.