Is the Federal Reserve issuing money in a fair way? Basic logic says that, once a central bank is issuing new money for increasing the money supply, the money should be distributed fairly between all the citizens, and that means giving them all an amount equal in proportion to their income (say a 5% of their annual income for achieving a 5% inflation). So they should give the money to the government, and then the  government either distributes the money or it builds public infrastructure that everyone benefits from. Most likely, this is the way the Eastern European countries experienced inflation in the 1990's.

Conversely, when the Fed decides to remove a certain amount of money from the economy, the Government should tax everyone in a fair way (the rich should pay more than the poor) and then give the money to the Fed to burn it. Or to burn it directly, without even giving it to the Fed.

Instead, the Federal Reserve has some sophisticated and arcane ways to move the money, beyond the comprehension of the general public. The Fed and the banks are involved into actually owning those newly issued money, and that looks really strange.

This article says:


  When the Fed wants to expand the money supply, it buys a security -- let's call it Asset A -- from a bank. Then it electronically transfers money to that bank.


Therefore, if the Federal Reserve decides, it can issue new money and buy and own half of the assets in the USA for free whenever they want. The newly issued money get to the banks, and depending on the decision of the banks to use the money or not, they can create inflation - or not. They can give those money in astronomical bonuses to their top executives and those people can send the money to Switzerland for example - therefore no inflation.

Is there something important that I'm missing?
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.