Enemy combatants should be tried in the United States court system if they are citizens of the United States. A person's constitutional rights do not go away simply because that person has been accused of being an enemy combatant. An accused is still entitled to due process, an attorney, and a trial by jury.
A foreign national acting as an enemy combatant on foreign soil should be subject to military tribunal as per the UN Charter and other International Law. A US citizen acting as an enemy combatant on foreign soil should also be under tribunal. Only acts on US soil should go to civilian courts. Even here foreign nationals may be subject to tribunals.
No, they should not. Enemy combatants are not (usually) citizens of the US, and therefore should not be afforded the same framework of due process afforded to citizens. That is not to say they should be treated inhumanely, however. They should be tried in Military Court such as the United States Military Department of Justice, or an international consortium of judges.