You kind of buried the most serious thing -- he is hitting. How much? Which kids? Or adults only? What is the situation when he hits -- is it when he has to switch activities and doesn't want to? Is it when he is tired near the end of the day? You need to have a LONG and detailed talk with not just the director but every classroom teacher who deals with him. Hitting can get a child kicked out of a daycare or preschool and you need to know when and under what circumstances he seems likeliest to hit.
What is the daycare's reaction? How do they intervene? Do they make him apologize to the person who was hit and then give him this "cozy corner" which he will not connect with discipline so it won't change his actions? Has the daycare learned to foresee when he's in a situation where he's likely to hit, so they can remove him from those situations and stop it before it starts?
Please focus on the hitting first. He must get past that and he won't if they just gently say, "Oh, don't" and "go to the cozy corner" which is likely a reward in his mind.
Yes, your son should have been corrected swiftly enough that he didn't get as far as standing on a table, which makes me wonder how closely supervised the kids are in the nap time. Is this time also considered "break time" for some of the staff? They may have too few adults in the room.
By his age, most kids are done with naps, period. I am surprised they expect the kids his age to be still and quiet for two solid hours. Does this day care have a mixed-age group, where your son and children his age are expected to have this long nap period because younger kids, who do still need naps, are napping at that time? The daycare should have enough staff, and enough facilities, that it can separate these older kids and have them doing something much more constructive during the younger kids' nap times.
I might reconsider the daycare if they are lumping all age groups together. He should spend his time with peers around the same age.
Please tell the director you're "taking back" your suggestion that she "keep him in" from outdoor play as discipline. This would only make him much MORE likely to act up, be restless and climb around! He needs to be burning off energy, not being held down in the name of an enforce nap time that does not really apply to him. If he would not be napping each day at home, why would he nap at daycare where things are so interesting and there are so many kids and adults to react to him?
He also is getting used to following directions from adults who are not you or his dad. That is a critical skill for a child to learn before starting kindergarten, so it's better for him to act out and get corrected, and learn to function in a group, now -- rather than when he's five and in K.
I really would get him into a daycare where he is not expected to nap, and there is much more proactive provision of things for kids his age to be doing. You say he's very active so this may be a daycare that is too mellow for him, or has too many kids of mixed ages all together.
Depending on your work schedules and needs, is there any way he can go to a preschool, maybe four days a week for three hours or so each day, rather than a full-time day care? He may do better in a more structured setting where kids are engaged every minute because they're in a preschool curriculum, not just a daycare. Some daycares claim they do "preschool" but they don't come near what a preschool really offers. Look for a place where the staff are trained in child education and development.