Answer: Primarily the U.S. District Courts, though there are other trial courts as well, including magistrate courts, the bankruptcy courts, the U.S. Court of International Trade, the U.S. Court of Federal Claims, various military courts, and the U.S. Tax Court.
For a detailed summary of the U.S. federal court system, see "Understanding Federal and State Courts" at <http://www.uscourts.gov/EducationalResources/FederalCourtBasics/CourtStructure/UnderstandingFederalAndStateCourts.aspx>
Answer: The US District Courts hold 80% of federal trials; the US Court of International Trade and US Special Courts, combined, hold the other 20%. Cases of general jurisdiction enter the federal judiciary through the US District Courts.
Answer: No. The federal courts are part of the Judicial branch of government, which is co-equal to, and independent from, the Executive branch (the US President). The President has no authority over federal courts, except for having the power to nominate federal judges and US Supreme Court justices when vacancies arise during his term of office.
The President has no authority whatsoever over state courts.
Answer: It depends on the matter and whether it is a civil matter or a criminal matter.
Added: There are areas where state and federal jurisdiction overlap, however - one of them would have to surrender their jurisdiction to the other. The case can not be heard concurrently, only one system can hear or try the case.
Both the state and federal court systems have appellate courts that review cases that were originally tried in a lower court. Examples of federal appellate courts are the US Court of Appeals Circuit Courts and the Supreme Court of the United States.
Answer: It depends on what you are referring to. An order of the civil court cannot overrule a finding of the criminal division any more than a finding of the criminal division can overturn a finding of the civil division. However a ruling of one may have direct effect on the actions of one of the principals in a case in the other branch.
Answer: Inferior courts refer to trial courts (or courts of original jurisdiction), as opposed to appellate courts, in both the state or federal court systems. In the federal court system, the inferior courts typically refer to US District Courts, but may also refer to special courts, such as the US Court of Federal Claims, as well as territorial courts that function as district courts for US territories such as the US Virgin Islands, Guam, etc.
Answer: No. Equity Courts were designed as a separate court system that takes care of cases that do not fall under common law. Any case that could be heard in an equity court could be heard in a law court but not vice versa. Equity courts are for situations where a judge can listen to the facts before them and make a decision then and there. Lawyers are not allowed to present cases which would show how precedents apply as they do in a court of law. Thus, wills are probated. Warrants are signed. Bankruptcies are processed. Even lawsuits for small amounts of money are settled in a few minutes. One thing after another happens like clockwork. Nothing comes before the court that would cause anyone to go to jail. Nothing comes before the court that would cause anyone to lose a large amount of money.
Answer: In the Federal Court System, the US District Courts ARE the lowest level of court, and have original jurisdiction (over FEDERAL offenses) within their assigned districts (of which, I believe, there are 94).