Yes. Yes, he is. That's what presidents do. And since it is just what presidents do, you need to calm down and accept that this is basically just normal politics.
In December of 2006, we saw something rather different from normal politics. In the middle of George W. Bush's second term, he selectively fired a set of US Attorneys for their failure to prosecute cases of voter fraud. The Bush administration had told them to go after voter fraud-- a longtime Republican boogeyman-- but since voter fraud doesn't exist on a large scale, and even individual cases are incredibly rare and hard to find, the Attorneys couldn't do it. So, Bush fired them. That was more of a violation of norms.
This isn't. Trump is behaving normally and within the bounds of what presidents are supposed and entitled to do. Reports have come in that he did it in a disorganized and douche-y way because he's Donald Trump, but focus on substance, not style. This is normal.
However, it is important to remember the December, 2006 incident. Voter fraud. Trump is big on this nonsense. If he tells people to go after voter fraud, one of two things will happen. Case 1) bullshit prosecutions that fall apart, or Case 2) failure to prosecute, and we revisit 2006. Which do we see? It depends on who he hires.
But, that's the future. The firings? Normal. Get out of your bubble.