Do the judges of the higher court have the authority to request to
review the legal case that happened in the lower court, such to review
the court case and make a decision on the legitimacy of the rulings
that happened in the legal case?
No.
Many appellate court have administrative or supervisory authority over lower courts that are subject to them that allow them to promulgate court rules and administrative procedures (e.g. declaring courts in some or all of a state closed or subject to restricted hours due to a natural disaster or pandemic). But, this is something very different from a ruling on "rulings that happened in the legal case" in a court over which it has appellate authority.
Probably the greatest extent of the administrative power of the courts to act sua sponte is the authority of the appellate courts to consolidate related cases pending in multiple trial courts with a single trial court judge. Somebody has to petition to do this, but there doesn't need to be a petitioner requesting this in each of the cases to be consolidated.
In some extraordinary circumstances, a court that finds a party or viewpoint to be unrepresented, for example, when the state attorney-general refuses to defend the constitutionality of a state statute, can appoint counsel to present the views of that party or viewpoint to the court making the appointment.
Another prominent example of this being down is the Amistad case involving a slave ship in which the slaves had risen up, taken control of the ship and landed in a U.S. port, in which the court appointed an attorney to represent the mutinying slaves who did not speak English or have any understanding of the U.S. legal system or any funds of their own.
Another distinction is that some U.S. appellate courts have a limited original jurisdiction. Thus, even though the court mostly handles appeals from lower courts, there are a handful of matters which can be initiated in the first instance in what is usually an appellate court. For example, the U.S. Supreme Court has original jurisdictions over lawsuits between U.S. states, and can issue some writs based upon a petition filed in the first instance in the U.S. Supreme Court rather than in a lower court. More prosaically, the Colorado Court of Appeals can resolve some issues related to costs and attorney fees incurred in an appeal in the first instance in the appellate court. And, many state supreme courts have original jurisdiction over attorney and judge disciplinary proceedings, which a regulatory official in the judicial branch (who is not a judge) can initiate.
But, unlike, for example, the higher courts of France, which routinely audit a small percentage of trial court rulings, a U.S. appellate court cannot sua sponte (i.e. of its own accord) initiate review of a lower court ruling. This is a natural consequence of the "adversarial system" in the U.S. court system. Somebody has to petition a U.S. appellate court to do something before it can act with respect to proceedings in another court over which it has authority.
I would also note that there isn't necessarily an outright jurisdictional prohibition on such a review, although, in federal courts, the "case or controversy" requirement of Article III of the U.S. Constitution does impose some limitations.
For example, Congress passed a law a couple of decades ago that mandated the auditing of a random sample of bankruptcy cases. But, since this is conducted by an official other than a judge who can serve as a party seeking review of inaccuracies in a case or criminal fraud prosecutions, bankruptcy audits do not offend the "case or controversy" requirement, or the notion of an adversarial system in which appellate courts are purely reactive.
