There are a few of these trailers around, even some that are made in the US, including some from a company that sells multiple options (although they may have stopped, everything is out of stock on Amazon and not on their website). Most are DIY, and some have fairly detailed instructions available.
This how to make a cargo trailer question probably answers half of your question, so I'm going to focus on the other parts.
It's worth spending a lot of time reading and researching, and I suggest contacting people who have trailers like this and asking for their opinions. If you're lucky someone will give you one (admittedly in the "if you come and pick it up" sense), but that could be a good way to start your first camping holiday with it. Travel to the trailer with your bike, grab the trailer, then ride home. Sure, if it turns out to be a dumb idea you may want to abandon it after a few thousand kilometres, but that's cheaper and easier than building your own then discovering you don't like it.
I also suggest that you buy a standard bicycle cargo trailer first as they're much, much cheaper than any camper trailer that you buy or build. Tow that around for a bit, use it for your grocery shopping, take it camping. Think of it as a cheap testbed for your ideas about bike camping. Because it is. If you don't like towing the trailer, you find you can't easily tow even the 30-50kg those trailers are rated for, or you discover other problems, you won't have wasted much time or money on the camping trailer.
On that note, watch for the sunk cost fallacy, with other buyers and especially with DIY builders. DIY has an awful lot of "I built this, I put in a lot of work, it must be a great thing that I have built".
The Danish one looks as though it's rotomoulded, purely because getting something that big injection moulded would mean a million dollar mould. If you want that sort of custom plastic work done, it's going to be a lot cheaper to import one from Denmark than build it yourself. Even if you own a rotomoulding factory.
That means building it some other way. Corrugated plastic is easy to work with, fairly strong, resilient and cheap, and it also insulates more than nylon tent material does. But the easy way to do this is the buy a tent and put it on a trailer - that's what the "Kwik-Kamp" ones do. Heavier, but not much better, would be sheet metal, probably aluminium because it's easier to make things lightweight with it (steel can be just as light, but that takes more skill).
The choice really comes down to what you have available, and what you're comfortable working with. Being able to weld steel tubing will make the whole process much easier, even if you use plastic or wood as your sheet material. A steel chassis and hitch will be simpler than trying to build from other materials.
I would start with a simple flat two wheeled trailer that can carry your camping gear. Go camping with that and see how you like towing a trailer. Then add 50kg of water to it (because water is cheap and easy to get), and go again. If you're not comfortable towing 100kg, the camper trailer is not for you.
Then I'd build a basic, big, trailer probably like my "megatrailer"
That was only 1.5m long, but it would be pretty easy to make one 2m long, enough to fit a narrow mattress on. You could even make a divan base for it to carry your stuff, or just a wooden base and a box on top, then strap the mattress on top of your stuff. As long as it doesn't rain. But this is a prototype. Maybe also do one trip with a fridge or oven box on top of the prototype, to give you an idea of how wind resistance will change before you get too carried away with building.
Once that works I'd add the real "camping box" to the design. I would be tempted to make a complex folding version just to keep the towed size down. Hinge across the back or front to halve the length, then tilt the front wall back to reduce air resistance, and if possible tilt the sides in too. Having flat walls that are trapezoidal, so the erected shape is a truncated rectangular pyramid, might make it possible to fold the walls completely flat while travelling.
One variant of that would be to find a tent that is the right size and shape for your trailer, build a topless box base that's the size of the tent, then glue the bottom of the tent to the base and cut the floor out. That way you have a fairly waterproof cover for your box, and it should be easier to set the tent up than without the trailer. Downside is you're still sleeping in a tent.
I would be tempted to push the "box" up a bit, possibly to around a metre. That way you have enough structure to lean on the fixed wall while you're sitting in it, and with a bit of design effort it could still be fairly aerodynamic. You could possibly do a "pop top" style camper based on that, or the more modern wall lift variant - two boxes that nest, so when you lift the outer one that is the roof, it exposes more and more of the inner walls until eventually you reach the join and it's fully erect. That gives you completely solid walls and potentially a very weather-resistant design, but obviously at a significant weight penalty.
Another option would be to copy the A frame style pop top campers
That has the advantage that you know the design works, but it would need to be fairly large to give you enough headroom. But possibly you could vary that design so that it worked better in small scale.
There's a good reason why the Wide Path photos show 406 wheels all round. Trailer wheels, and the rear wheel of the towing bike, experience significantly more side forces than most bike wheels. You will need to take that into account. I use 406 wheels for most of my bikes for this reason, and also partly because it's a good idea to pick a size and stick with it. Having to carry two sizes of spare tyres and tubes while touring is just annoying.
To get the lateral strength you can either make the wheel smaller (406 size), the flanges bigger or the hub wider. Fatbikes, IGH and motorised hubs are other solutions to that problem. You can, of course, combine them - a fatbike with 406 wheels and hub motors might barely have spokes at all, so getting strong wheels would be easy (lacing them up, maybe not so much)
One option would be to design it around 300-500W of solar panels, and use an electric bike to tow it. That way you can go further, faster with the trailer and as a free bonus you have power when you're camping (not much, but charging phones etc will be easy, and you'll be able to run a light).
I met a guy who is riding around remote bits of Australia using a fatbike and a trailer load of panels. He uses a tent, but the idea is there.
Travel Limitations As you've already realised, you're not going to be taking this on a plane or bus. Ever.
Size is going to be an issue in other places too, unless you already know everywhere you want to take it and know that there will be metre-wide gaps everywhere. Unfortunately many "bicycle facilities" have barriers specifically to stop vehicles that are wider or longer than a normal bicycle from using the facility. So if your design is wider than the usual 800mm or so bicycle, you either need to be able to lift it over a 1.5m high fence, or roll it backwards until you can find somewhere to turn round.
The fact that most camping trailers can't do that makes me wonder how serious the owners are about cycling - I suspect that most owners ride short distances on flat ground on known roads. If that's what you want, this could work really well.
Other Random Observations
I've done a bit of cycle touring on odd bikes, including a tandem recumbent trike, and on one trip I had both the trike and megatrailer. We quickly discovered that with the trailer our speed dropped from about 20kph to about 10-12kph, and the other cyclists were generally no faster with the trailer (they were usually faster than the trike without it). Touring with a big, heavy bike starts to get tedious after a while.
Sam, the solar trailer guy above, started out with a home-made tadpole solar power-assisted trike. After a few thousand kilometres he decided that he really, really wanted something that could detach from the solar panels and be ridden independently. Your camping trailer will have that advantage, but it's worth keeping in mind that you will want to lock and leave the trailer at times. Resist the urge to build a heavily customised towing bike that is hard to ride without power assist.
But you may well find that you're better off with a custom towing bike. Lacing narrow-ish rims onto fatbike hubs will give you unreasonably strong wheels, and with the camper putting big side loads on your rear wheel that may be really useful. You might also find a Pinion bottom bracket gearing setup solves a lot of problems around where you attach the trailer, and how you get the extra-wide range of gears you will hopefully need (if you're unlucky it will handle so badly that typical MTB gears, only with unusually small chainrings, is ample for the 0..20kph speeds your rig is capable of).
Designing it to be dismantle-able into bits that can be shipped might be well worth while. Not so much "easy to reassemble" but just "break it down, ship it home, get on the bus, worry about it when I get there" scenarios. You'll possibly only do that once, but if you do need to bail on a ride not needing to hire a box trailer or truck to move it may greatly simplify things. It's better than just setting fire to it, however good that might feel at the time.
Finally, I do think this could be a really good project, but I would suggest testing your "tow a big heavy trailer on tour" abilities first before committing too far to the idea.