Appropriate statistical test
Regarding your second question:


  The users who did not have and tweet for the time range, is this okay to assign difference in mean zero, or I should exclude them from the samples?


You essentially have missing data in this case. How you can deal with this will depend on the model you are using, and if it is robust to missing data. If the model can ignore $\mu = 0, \sigma = 0$ values, then try it out. Otherwise, you might want to leave them out as you suggest, or perhaps even impute them with their previous known values.
If you are e.g. using something like an ARIMA model, then it keeps track of a moving average. In this case, using zero values will have an undesired impact (assuming zeros are not common in general).



I'm not sure I understand what you are asking in your first question. What have you tried already? Have you got some results?