What safeguards are built into the Federal Appeals Court system to ensure that decisions are made properly

1-What safeguards are built into the Federal Appeals Court system to ensure that decisions are made properly?

2-What is mootness? Why is mootness a disqualifying factor in a case coming before the Supreme Court? What does precedent have to do with mootness?

3-How do outside groups and individuals without standing in a case attempt to interject themselves into the hearing of cases by the Supreme Court?

4-Are the federal courts ‘imperial’, or are they subordinate to elected branches of government? In what respect does the federal judiciary still pay a ‘checks and balances’ role?