Yes, you can compute a single-bin DFT at ANY frequency, as mentioned in this post, and a related post. After all, F = k*f_s/N, where f_s is the sampling frequency and N is the number of points. Since f_s and N are known, just compute the value of k you need to get the frequency you want. It's basically putting down a single bin DFT centered at frequency F. And the 'bin width' is the same as what you'd get with any other DFT bin based on N points.
Since the advent of FFT, people have become accustomed to thinking of the parameter 'k' (the frequency index) as being an integer. However, if you think of a DFT calculation as being like a correlation, then you can think of it as answering the question: does my time record look like a sine or cosine wave of a specific frequency? So, no matter what the value of 'k' is, and no matter if you have an integer number of cycles in your input or not - for example, 3.7825 cycles in your input will correlate quite well with 3.7825 cycles of the sine and cosine wave of your DFT bin. You just have to figure out what value of 'k' to use.
As for phase - well, whether for a full DFT, single-bin DFT or FFT - you've DEFINED the time = 0 point (you have a data point x(0), don't you?), and phase will be relative to that point.