I've heard a DA can choose to prosecute or not to prosecute people for virtually any reasons.
The concept is called prosecutorial discretion. It is the authority of the district attorney for each district (and ultimately their attorney general) to choose whether or not to press charges in a case, offer a plea bargain or other alternate sentencing arrangement, or decline to prosecute. As a practical matter, not all cases have enough evidence to convict and the district attorney's office would prefer in many cases to get a plea bargain with a lesser sentence than have to spend the time and resources going through a full court case.
What would stop a DA from, say, not prosecuting his own family? What would stop a DA from not prosecuting presidents?
A district attorney should recuse themselves if a case is personal to them, and either leave the decision in the hands of a neutral assistant district attorney or escalate it to the attorney general. That's not to say that their relationship with a suspect can't cause them to get treated differently anyway, but that's just a fact of life. Since district attorneys are usually voted in such behavior can cause them to lose the next election.
Prosecuting the President is a more complicated topic, but the President has no general statutory immunity to prosecution. The federal Justice Department's current policy is not to press charges against a sitting President, mainly because it would severely undermine their authority and such a drastic move is thought to be better left in the hands of Congress, who can impeach a President in order to have a trial which can result in removing them from office. On the other hand, some commenters have suggested state district attorneys could file charges against the President, and are most likely within their rights to do so. Considering how many other questions have talked about the possible effects of this, I'm going to just leave it there.
I've heard justice department is under the executive branch
The Justice Department is a part of the Executive Branch in the US and is headed by the Attorney General, who is a member of the President's Cabinet (states may have this structure or may have their Attorney General as a separate entity to the governor, and most actually elect their Attorney General rather than appoint them). The idea is that the Executive Branch is responsible for enforcing laws, so the Justice Department should be a part of it. The Attorney General is unique among Cabinet members in that they are considered to have a "special independence" from the President to practice their discretion as they see fit.
I've also heard that judges and often presidents can make laws too.
Not exactly. Only the legislative branch can make statutory laws, but the way other branches interpret and enforce those laws also have force of law as long as they are compliant with statute (except in the specific case of judicial review, where the courts can override the legislature). The President signs "Executive Orders" which outline how the Executive Branch is going to execute laws passed by Congress. Laws usually give the Executive Branch a goal to pursue and money to do it, but leave most of the specifics up to the actual agency executing the law - obviously lawmakers can't account for every individual situation under which the law is being applied. Some Executive Orders have been criticized for stretching a law further than it was intended, and some are criticized for implementing laws in such a way that it undermines the law itself (e.g. by using the funding in a way which is technically conformant but leaves little or no funding for an important objective of the law).
The Judicial Branch (not to be confused with the Justice Department) is ultimately the arbiter of law enforcement (referring to the "enforcement of laws" in general, not just criminal law). The US uses a "common law" system, a part of which is the importance of previous case rulings ("precedent") in current cases. This means that some cases may "make" law by becoming the precedent cited in future cases for how to apply the law to a common fact pattern. Precedent can be overruled by new laws, and can also be tricky to apply since no two cases are exactly the same - sometimes there is some part of the fact pattern of a case that causes the result of the case to be different than precedential cases.
The courts can also choose not to allow enforcement of a particular law if they find that it violates a part of the Constitution or a person's rights. In these cases (which usually involve criminal law but can involve civil law), appeals can be heard by district appeals courts and eventually the Supreme Court. The Supreme Court is the ultimate authority on whether or not a particular law violates the Constitution (and how a law should be interpreted in general), and have the authority to make such rulings on both federal and state laws (which is unique from most of the rest of the government, where the federal government's structure and laws are separate from all state government structure and laws). This is probably the closest process to "making" law outside of the legislative branch - if the Supreme Court rules that a particular law violates the Constitution, the legislative branch has no direct recourse and must either change the law or accept that it will not be enforced (they can also begin the process of passing a constitutional amendment, but that involves the legislatures of the states as well).
Is this what's happening in US too?
I don't know much about Indonesian law or Mr. Prabowo, but false claims made in general are not a violation of law in the US. False claims about a person can be defamatory, but there exist almost no criminal defamation statutes - the person defamed can just be awarded damages and the defamer can be ordered to stop (which it would be a crime to ignore). Most precedent in this case has to do with the First Amendment's guarantee of Free Speech.