Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized function? As for measure, concentrated at point $0$ is it possible to say that its density is $\delta(x)$?
Question 2. If the answer to first question is positive, is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to $k$-dimensional Hausdorff measure is always defined but may be the generalized function? Isn't it the same as $k$-density of measure? (Krantz, Parks, Geometric Integration Theory)
A1. A generalized function (a distribution) is a continuous linear functional on the space of smooth compactly supported functions endowed with appropriate topology. Under this definition, every locally finite Borel measure is a generalized function. But I would not want to use the words "Radon-Nikodým derivative" here, because nothing resembling a derivative is taken. I interpret the Radon-Nikodým derivative as the density of the absolutely continuous part of a measure. That is, I would say that the RN derivative of $\delta_0$ is identically zero, but this measure has a nonzero singular component.
A2. You can consider upper and lower $k$-dimensional densities. I would not call them Radon-Nikodým derivatives, because they do not recover the measure by integration, not even its absolutely continuous part. For example, on $\mathbb R^2$ the measure $\mathcal{H}^2$ is absolutely continuous with respect to $\mathcal{H}^1$. The upper and lower $1$-dimensional densities are zero: $$\lim_{r\to 0}\frac{\mathcal{H}^2(B(x,r))}{\mathcal{H}^1(B(x,r))}=0\qquad \forall x$$ But integration of $0$ with respect to $\mathcal{H}^1$ does not reproduce $\mathcal{H}^2$.
But in general, the answer to "is it possible to say" is always yes. It's possible to say the silliest things. I could define the Radon-Nikodým derivative of $\mu$ with respect to $\mathcal{H}^k$ as $$ \frac{d\mu}{d\mathcal{H}^k}(x) = \begin{cases} \lim_{r\to 0}\, \mu(B(x,r))/\mathcal{H}^k(B(x,r))\quad &\text{if the limit exists} \\ 42 \quad &\text{if the limit does not exist} \end{cases} $$ The more focused question would be "is it possible to define $\frac{d\mu}{d\mathcal{H}^k}$ so that it has the properties P1, P2, and P3?"