When the US Supreme Court declines to take a case, can they try again right away?

If the SCOTUS does not take up a case.it is implicit that the lower court’s ruling was correct and there is no outstanding question of law.

The Court may refer a case back to the lower court because the lower opinion was incomplete. In those instances, the Court may review the case again at a later date if a point of law is in conflict with another circuit, or if the appellant raises a question of error.

The SC is the final court of appeal.
See above.
If they do not take on the case it is because they do not think there is a reason to debate the lower court decision.