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.

Thank you all for helpful replies! Here is a DRAFT itinerary. More suggestions welcome!

FRIDAY, SEPT. 19
--9:55 a.m. arrive Copenhagen CPH
--Visit Copenhagen
OVERNIGHT: Copenhagen hotel TBD (but near Copenhagen Central train station)

SATURDAY, SEPT. 20
--Visit Copenhagen
OVERNIGHT: Copenhagen hotel

SUNDAY, SEPT. 21
--Visit Copenhagen
OVERNIGHT: Copenhagen

MONDAY, SEPT. 22
--Morning train from Copenhagen Central Station to Hjørring St
--Mid-afternoon pickup at train station by Family 1
--Visit with Family 1 (Tårs)
OVERNIGHT: Family 1

TUESDAY, SEPT. 23
--Family 1 activities in Northern Jutland (Skagen, family farm, etc.)
OVERNIGHT: Family 1

WEDNESDAY, SEPT. 24
OVERNIGHT: Family 1

THURSDAY, SEPT. 25
In the morning, by car:
-- Family 1 car transport from Tårs to Family 2 in Tolne Kirkby (20 min.); Farewell Family 1!
-- Visit Family 2
OVERNIGHT: Family 2

FRIDAY, SEPT. 26
--Family activities in Northern Jutland
OVERNIGHT: Family 2

SATURDAY, SEPT. 27
--Morning transport by Family 2 to Hjørring St
--Morning train from Hjørring St to Aarhus Station (10:04-12:08)
--Farewell Family 2!
--On our own to visit Aarhus
OVERNIGHT 10: Aarhus hotel TB (but preferably near train station). Suggestions?

SUNDAY, SEPT. 28
--Explore Aarhus
OVERNIGHT 11: Aarhus hotel TBD

(The last two nights are TBD. Want to see the sights in Aarhus so staying there at a hotel near the train station seems like Option A. Option B would be to visit Roskilde and maybe Odense from Copenhagen as a day trip. Suggestions?)

MONDAY, SEPT. 29
Somehow get from Aarhus hotel to Aarhus St.
--Morning train from Aarhus St to Roskilde St OR to Copenhagen H
--Explore Roskilde, including Viking Museum to visit ancient relatives -:)
OVERNIGHT 12: Roskilde hotel near the train station OR Copenhagen hotel near train station

TUESDAY, SEPT. 30
--Explore Roskilde and/or Odense and/or more of Copenhagen
OVERNIGHT 13: Roskilde hotel near the train station OR Copenhagen hotel near train station

WEDNESDAY, OCT. 1
--If in Roskilde, somehow get rom Roskilde hotel to Roskilde St; arriving by 8 a.m. to buy train tickets to CPH Lufthavn
--Train from Roskilde St. to CPH Lufthavn:
o 8:20 Inter City IC 208 direct arrives 8:53 a.m. after 4 stops
o 9:20 Inter City IC 214 direct arrives 9:53 a.m. after 4 stops
--If in Copenhagen, train from train station to CPH Lufthaven, arriving by 9:30 a.m.
FLIGHT HOME DEPARTS AT 12:30 P.M.

Well done, you! You saved yourself piles of money by staying with family and not renting a car! And your trip sounds awesome! I love Aarhus!!! I don't think it's considered a tourist attraction, but I encourage a visit to DOKK-1, which is a government building that houses a library, makerspace, indoor and outdoor children's play areas, some sports facilities (I think), a cafe, and public-facing government agencies like public benefits, licensing, etc. And it has an automated parking garage, which is cool to see in action.
Also great are the modern art museum with its rainbow bridge, the Moesgaard Museum, which has a bog body, and the infinity bridge. Public transport is good, so don't worry about getting around.
Have a great trip!

I suggest you stay one or two nights in Odense to visit the Railway museum, The Funen Village, The Hans Christian Andersen Museum, HCA's House and other things.

Also just walk around. Odense is the third largest city in Denmark, but the center is rather small and concentrated. I love sitting at a cafe by the "river" with a glas of coffee, wine or beer.

That looks like a pretty good itinerary!

Aarhus is a very nice city in my opinion. Can wholeheartedly recommend. In addition to Moesgaard Museum and Aros mentioned above (both well worth a visit), there is also Den Gamle By open-air museum, which is quite popular - and that is well deserved. These can probably be called the 3 big attractions of Aarhus. Of course Aarhus has a lot more to offer. One example being the Gender Museum KØN - and while it may not be to everyone’s liking, there is a lot of interesting and thought provoking stuff in there.

Regarding your last two nights:

There are good things to say about both Odense and Roskilde. So I do not think there is a wrong choice. Odense is much bigger and was home to H. C. Andersen, so if you are interested in that, then you should be visiting H. C. Andersen’s House - which have a lot more to offer than “just” his childhood home. It looks like you’d do Odense as a day-trip from Copenhagen? That is possible, but you’ll be doing a lot of back-tracking. I suggest that you on your return from Aarhus on the 29th stop in Odense and stay the night there.

Roskilde, while being smaller, has a cozy town centre and some interesting (and historically significant) things to see. You mention the Viking Ship Museum (if interested, you can book a sailing trip on a replica viking ship on their website). There is also the cathedral in Roskilde, which is the resting place of Danish kings and queens - which is the main attraction when visiting the cathedral. But if you choose Roskilde, consider taking the train there from Aarhus in Monday afternoon instead, since it is smaller and those extra hours are probably better spent in Aarhus.

DOKK-1, which is a government building that houses a library, makerspace, indoor and outdoor children's play areas, some sports facilities (I think), a cafe, and public-facing government agencies like public benefits, licensing, etc.

The bundling of public services in libraries is quite common nowadays, but Dokk1 is a bit of a crown jewel at that (at least here in Denmark). And it also houses the weather studio of the Danish Broadcasting Corporation. In that sense, I visit there several times a week ;-)

Does anyone have any gift suggestions for Danish relatives? In a token way, we'd like to demonstrate gratitude to our hosts. Whatever we take must fit inside a carry-on suitcase (no checked luggage for us). One relative owns a classic Harley motorcycle so I've considered a Harley keychain for him. Many Euro women seem to wear scarves but who knows what color or style they might like?

FYI, our primary hosts are retired and do not know each other. Both will be inviting their offspring for short visits while we're around. One has a daughter who's taking time off from work to help shepherd us around Northern Jutland. Separately, of course, both primary hosts visited us several years ago (including the daughter) so I'm pretty sure they view our visit as kind of a payback.

FYI, one family has suggested we go out for dinner so we'll insist on picking up up the tab for that. But still ...

Definitely take them out for a meal. I generally recommend something from your local area. Since they have visited you before, that may not be as applicable. That said, local artisan chocolate is always a good choice. If your area is known for something in particular, bring that.
A scarf is a nice idea for the daughter who will be driving you around. Something locally made like a hand-painted silk or hand-knit, lightweight wool or a silk blend. As far as colors, go for all of them or none of them (meaning several colors or shades of black and grey). Maybe ask for your wife's help in choosing something pretty?

Just realized a bottle of aquavit, purchased locally, saves space and works well as an arrival gift!