Dear

1. Actually the capacity R = log2(1+‖ vHw ‖^2) is the same as C=log2(1+SNR). The term ‖ vHw ‖^2 is nothing but the signal power from the precoder side to the decoder side. It seems the authors of the attached paper made the assumption that the noise power is unity. Hence, SNR = ‖ vHw ‖^2/1.

2. Yes you may define the Rayleigh fading as you described.

3. Please be more specific and describe what other distributions you want to generate.    

There is only one Shannon capacity, defined as the supremum of the mutual information between the input and output: https://en.wikipedia.org/wiki/Channel_capacity

However, the capacity expression takes different forms for different channels.

1. What you provide is not the capacity expression, strictly speaking. If you have a MIMO channel, the MIMO capacity requires to send multiple data streams and selecting the precoding based on the SVD of h. Here is a paper about that: https://infoscience.epfl.ch/record/95836/files/Capacity%20of%20multi-antenna%20Gaussian%20channels.pdf

Your expression can, however, be used as achievable rate that is smaller (or equal) to the capacity.

2. You can define the channel matrix like that, but you cannot use your expression to compute the capacity. One reason is that you formula does not provide the MIMO capacity. Another reason is that your formula is for deterministic channels and not random channels. The paper above covers also the capacity of fading channels.

3. The expression for H is fine. It is called i.i.d. Rayleigh fading. There are different forms of correlated Rayleigh fading as well, but I think it is important that you first learn the i.i.d. Rayleigh fading case. It is the simplest one.