I don't know why someone is there to boss over users' question. When the editor modifies the question, my question is: did he(she) ask this question or I did ask. I am totally shocked because the question is modified in such a way that the purpose of the question is missing and instead of getting wright answer, I will definitely come up with the answers that I already know. My question to the editor, will such modification continue in future. If the answer is yes, I am going to close my account right now. Thank you for bossing.

Hi,

i see that this is a rather old question, but since i came across it and found it as-of-yet unanswered, i decided to chime in. The approaches you describe have all been used widely and i suspect that Lorenzian broadening is in the lead as the most common. However, i should point out that your option #2 is really two different approaches: you can either use a Lorenzian function to broaden the energy levels and then either perform a summation over a large number of k points (essentially a zero-th order approximation to the integral over the k space IF the k space is continuous) OR perform a first order expansion of the energy around each k point and then use this linearization to perform the integral. 

However, you don't have to replace the delta function by a Lorenzian line shape in order to perform the latter type of integration--the integral of the delta function over k space is mathematically equivalent to the integral of 1/|grad E| over the constant energy surface. If the gradient is known (or calculated along with the E(k) dispersion), then the whole k space can be discretized and the constant energy surface can be well approximated by a plane perpendicular to the gradient within each discretization cell. This method was derived/proposed by Gilat and Raubenheimer in the 70s and remains popular today for bulk/3D materials. In 2D, the equations are different because the constant energy surface is no longer a surface but rather a contour (so you have lines instead of planes in 2D) but that makes the math easier. 

This method can be used directly without broadening OR combined with the Lorenzian lineshape (although this is actually more rarely done). I would say, since broadening tends to smear our the sharp features in the DOS, the Lorenzian line shape is used in the 0'th order simple summation over many k points where DOS(E)=sum_{all k} 1/pi Gamma/(Gamma^2 + (E - E(k))^2). I may be missing a constant to account for how much volume of the k space each discretized k point represents (which is needed to get the right units). Gamma is the broadening, often taken to be approximately k_B*T (the thermal energy), but more sophisticated approaches might relate the broadening to the lifetime of each energy level.