If an officer, warrant officer, or NCO received an award, can the same action the award was given be placed on their next evaluation?

If it is relevant to the area of the report and in the rated period, then yes. An Evaluation is not a military award. You can't recycle something and put it on more than one award unless it is for retirement and you do the ten year slew

Yes, as long as the action falls within the qualifying rating period.

If the action deserved an award then it should certainly be included in the evaluation.