Is the Federal Reserve issuing money in a fair way?
Normative questions require a standard of fairness in order to then be analyzed and answered with some degree of "objectiveness" so that the answer is not just a declaration of philosophical-ideological positions.

The OP starts by writing


  "Basic logic says that, once a central bank is issuing new money for increasing the money supply, the money should be distributed equally
  between all the citizens."


Why "basic logic" says that? Whose "basic logic"? I note that throughout human history and especially in the western civilization in the last 400 years, "fairness" is almost exclusively interpreted as "proportionality", not flat equality (i.e. as "per capita tax" is considered not in accord with current societal values, so are "per capita transfers").

Anyway, under the given fairness standard, certainly "new money is not fairly distributed". 

Regarding how the creation of new money works, and how it affects the domestic or global economy (or not), which is an additional question, I believe there are many threads here that provide a lot of information on the matter. Selecting the tag "inflation" or "money supply" will bring up most of them.