What is the most important things to look for post cutout on the first inspection of the hive?
Bump.
That you have a laying queen and honey/pollen. If they have those, let them be for a couple more weeks.
PS. First check should be 7 or more days after extraction.
I wait ten days before I open the hive; then look for freshly laid eggs and a queen.
Most of the cutouts that I rehive, seem to be a bit testy if they are opened up sooner than ten days. Sometimes they will abscond if they are bothered too soon.
Brenda