Federal judges are generally culled from party members of the state judiciary (for the purposes of the discussion I’m assuming we’re talking about Article III lifetime appointment judges). If a judgeship becomes vacant, other party members begin looking for jurists (and sometimes plain old lawyers) with good qualifications and recommend them to the president. The judge I used to work for was a former Texas Court of Appeals Judge recommended to President Ronald Reagan by Senator Phil Gramm.
Both, really. You’re competing with other members of your own party for the president’s appointment (and Senate confirmation) to a seat on the federal bench created by Congress.
You’re filling a vacancy, so you have to go wherever that vacancy is if you’re appointed, but it’s not like joining the Army or the FBI where they might just send you anywhere. Part of your job will be interpreting the law of the state in which you’ve living and judging/practicing, so you’re probably going to be appointed to a bench close to where you already are, and you’ll know where you’re going ahead of time. This may change if you’re appointed to the U. S. Court of Appeals for the Federal Circuit, which hears specialized appeals like patents, or if you’re appointed to SCOTUS. In these cases you’re going to D.C.
To a certain extent, yes you do, and so do your law clerks.
A federal district judge is much like the trial court judge that hears felonies or high dollar negligence claims in your state court. There are juries where required by law just like any other court. As a trial court judge, it’s pretty much just them ruling by themselves. U.S. Circuit Court judges are appellate judges, so they generally hear cases with two other Circuit judges, or more if sitting en banc. In the U.S. Supreme Court, of course, you’ll be rule with eight others.
For Federal District Court openings, the President will usually allow a Senator from that state who is of the same party as the Administration to decide who should be nominated. If both Senators are of the President’s party, the more senior one gets to make the call. If neither is, the President may rely on a same-party senior House member from that state or big player in the state party.