Most federal cases begin in the district courts.
Yes, that is the level at which most federal cases are first introduced.
State court cases begin in trial courts. They graduate up to the state level if one of the litigants is not satisfied with the decision of the lower court.
In most cases I have read the civil cases heard at the Federal level are cases that directly question someone's rights and apply to a large number of people. The cases heard at a local level can also be promoted to Federal cases. For example if you have a property dispute with your neighbor it will not go to federal court; however, if you have a court that disputes whether its legal for you to carry a concealed weapon or practice free speech you could proceed to the Federal Court.
Most federal criminal cases are tried in US District Court.
Generally, federal criminal cases are tried in US District Court.
Federal.
Federal appellate cases
State courts hear far more cases than federal courts.
This is admittedly an educated guess and may be applicable only in the US: But I believe the answer would be Municipal Courts or County-level court systems probably hear the most cases due simply to the sheer number of misdemeanor and traffic cases that are heard each day.
Federal trial courts almost always have original jurisdiction in the federal system.
No, most cases that begin in state court do not end up in the US appellate court. State court decisions are typically appealed to the state appellate court and may only reach the US appellate court if a federal issue is involved and the case meets certain criteria for federal review.