My answers to your questions:

- Federal Reserve Bank (FRB) has shareholders from banking institutions

Federal Reserve Bank, in short Fed, not FRB, does NOT have shareholders.  However, they do have advisory board in which some of the larger financial institutions participate.

- US Government is borrowing money from FRB for all governmental works at a loan

US government issues treasury bonds and other financial instruments to raise funds when needed.  It does not borrow money from the Fed.

- US Govt is paying the FRB and making them more rich, most of the money is coming from public's taxes

This is just not true.

- Question is why doesn't US Govt make it's own money instead of borrowing it from a for-profit banking entity?

No government, even our almost defunct Nepal government, can print it's own money all willy-nilly.  This will immediately devalue the currency and impact the economy most severely.  US dollar is the de-facto currency of the world.  If the government starts to print without any accountability, then not only the US economy but the world economy will collapse.  It'll be like in Germany after World War I or in Argentina few years ago, just 100 times worse.