Simple questions related to dft
The actual frequency in Hertz corresponding to DFT index $i$ with a DFT length of $N$ is

$$f=\frac{i}{N}f_s,\quad i=0,1,\ldots N-1\tag{1}$$

where $f_s$ is the sampling frequency. So if indices are numbered starting from $0$ (and index number $19$ is indeed the $20^{th}$ frequency bin, as assumed in (1)), then your answer is correct.

As for part 2 of the question, the answer is also given by (1), so

$$f_s=\frac{Nf}{i}$$

which gives $f_s=289.7\text{ kHz}$ for the given values of $i$, $N$, and $f$.