How to read off power at different frequencies
I know my x-axis gives frequency in Hz.


That is not the case.

A DFT is a mathematical transform between discrete signals in time and frequency domain. Just as as the number of your sample doesn't give you time in seconds, the number of the DFT bin doesn't give you frequency in Hertz.

Instead, the number of the sample gives you time in "sampling intervals $T$", and thus, the bin number in the $N$-point DFT gives you frequency in "Nyquist rate fractions" $\frac {T^-1}N = \frac{f_{sample}}N$, whereby you need to pay attention – the DFT should be understood as something that shows you both negative and positive frequencies, so therange is $-\frac{f_{sample}}2$, to  $+\frac{f_{sample}}2$ (for odd and even lengths, take care of the boundaries when calculating what the frequencies of the DFT bases are). The zeroth DFT base is always the average/sum/DC of the signal.


  if possible in dBm.


Not possible, unless you know the proportionality between your digital input signal to physical power, ie. you've calibrated your measurement setup and know which power a digital $1+0j$ has. Then: Parseval's theorem is your friend.

Note that the DFT is nothing but a linear operator – a matrix that maps $\mathbb C^N\mapsto \mathbb C^N$; the question how to interpret complex frequency domain values is absolutely up to your understanding of your system. Sometimes, the values' magnitude squares can be interpreted as estimate for the Power Spectral Density at the frequency of the bin – at other times, that is not the case (especially if you don't know whether your signal is weak-sense stationary). 

Sometimes, the bin of a DFT of a system is an impulse (ask a physicist about the meaning of the Fourier transform for solid state electronics), sometimes it's something like a pseudo-angle (ask an antenna designer), sometimes, it's proportional to speed of something (ask a radar engineer), sometimes, it's really just that — a complex, $N$-dimensional vector, that has no physical significance.

I'd like to quoute:


  One can Fourier transform anything—often meaningfully.
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ~John Tukey