Help with 13-night itinerary to Denmark

The closer ferry port is Hirsthals, for the Hirstlhals to Kristiansand ferry

Yes, thank you on the ferry info. From Kristiansand it's still a train ride to Oslo, of course.

I regret not getting out to Skagen to see where the two seas meet at Grenen, so I would definitely recommend that. Requires a car, of course. And therein lies the rub. Will you get a car after a couple of days in Copenhagen and start making your way north to Jutland? Whatever mode of transport you choose, I would suggest making a list of sights you want to see and adventures you want to experience, figure out how long each will take, then map out the transportation links.
If you decide to go by car after Copenhagen, one option might be to return the car in Fredrikshavn and walk on the ferry to Gothenburg, from where you can get a train back to Copenhagen.
I wouldn't short-change the visit with the relatives. Also, get their advice re: what to see and do in their area. And don't forget to call in on the relatives from "a few years back." This will be the best introduction to Danish culture your son can get.
Suggested reading ahead of this trip: The Year of Living Danishly by Helen Russell. It's a memoir of her move to Denmark (from London) and an exploration of why Danes are often ranked as the happiest people in the world. Really enjoyable to read.

Requires a car, of course

Not really. You can take a train to Skagen station. Then it is just a 4 km. walk to grenen.

Or you can walk 3 km. to "Fyrvej 38V, 9990 Skagen" and take "The Sand Worm"

There are many ways to tie such an itinerary together. When you are interested in the history and culture, there are many things you can see and do in both Copenhagen and Denmark as a whole. In addition to what you can find on this website and in guidebooks, Visit Denmark can also be a good (and free) place to start your planning.

In addition to the Hirtshals - Kristiansand ferry route mentioned above, which is served by the competing ferry companies Colorline and Fjordline, you can also:

• Take a ferry Hirtshals - Larvik with Colorline (closer to Oslo than Kristiansand, though the ferry terminal in Larvik is further from the railway station, as far as I can see)
• Take a ferry Hirtshals - Stavanger - Bergen with Fjordline (overnight)
• Catch a flight Aalborg - Oslo with SAS (direct flights on most days)

And for good measure, you can also reach Frederikshavn by both train and buses for onwards connection to the Stena Line ferry to Gothenburg. However, depending on your desire to explore rural areas it may be preferable to have a car. Some rural(ish) areas have decent public transport, so it’s a bit hit and miss when it comes to that and depends on your specific itinerary.

Some excellent, helpful posts here. Will share a draft itinerary once we sort out a few things (more and more it looks like Norway will have to wait another day).

One lingering question involves car rental. I think we'll use a car to visit family in Jutland. Love the flexibility, and we're thinking of taking the advice to consider visiting other, previously visited (by me) relatives in Jutland. Assuming we stay in a hotel near Central Station, Copenhagen, what is the nearest car rental pickup location? Would that involve, say, taking the train back to the airport and picking up the car there or is there someplace more convenient for picking up the car and hitting the road?

You don't have to go to the airport to pick up a rental car. I suggest you decide your itinerary first, then determine when and where to pick up the car. For example, if your next stop is Aarhus (a fine city worth visiting, IMO), you might prefer to take the train there and pick up the car at the end of your visit.

Just got back from Denmark, it's delightful, and as far as driving internationally goes, very pleasant and only stressful in construction zones-but we didn't take the car into central Copenhagen, I doubt that's as easy. There's even free parking many places (er, parkering, in Danish)
I think Denmark does a good job integrating transit (train, bus, metro, ferry) and car into a versatile network to get around the whole country. Consider having a car for part of the time and using transit in the cities, or between them.

We enjoyed our time in Aarhus, it was a good base for central Jutland.

Don't discount renting electric. Denmark has put in a very extensive network of chargers all over the country and are continuing to add more. There is a wide range of electric to choose from. New car sales there are close to 50% electric, in fact.

We were seeing gas prices like 13.79dkk/liter, which is a bit over $8/gallon. The charging was also cheaper.

I second the recommendation to visit all the family branches while you can.

I was highly impressed by the Danish museums. The ones we made it to (always more places to see than time! ) had very modern approaches using technology in combination with their artifacts to give depth and context to the subject. Not boring, not even the ship models.