Questions about reproducing results of a paper on Communication Systems and Wireless Communication
Let's tackle this piece by piece:

Path loss

Your path loss model gives the path loss in dB. Convert to linear scale. You can directly read $\alpha$ and $G$ from the result. 

Haven't read that paper (paywalled), but it looks to me like your shadowing should be part of the fading  $\beta$. Make sure your channel model is compatible with the one used in the paper at all! There's no guarantee that different sources use channel models that are.

Power Spectral Density to Power

Well, assuming white noise (check that assumption is correct for this paper!), the noise power in a channel is of course bandwidth of that channel times noise power spectral density.


  should I do that in the dB or Watts units ? 


dB is not a unit!

Generally, the actual units don't matter, as long as you do all the calculations correctly, so it really doesn't matter whether you're working in W or (MeV per minute).

Your first question, however, points to you not being 100% familiar with what dB and dBm are. I strongly recommend reading the Wikipedia article on decibel if that is the case!