The person in the video was building a stationary home. It probably didn't survive the first wind storm or snow; they are delusional about natural forces. As cyanrarroll says, snow is heavy and this is what it does to buildings that were built properly and to code at the time!
There are 2 kinds of trailered buildings: Manufactured homes which are trailered once from factory to trailer park and never move again; and RVs which move regularly. I doubt this construction would survive a one-time move from factory to final resting site, which would be the equivalent of a wind storm.
Entitlements and liability
A static building not on wheels is subject to permitting and inspections and this will not pass muster.
An RV is subject to ANSI/RVIA standards and regulated by NHTSA. Trailer frames are subject to DOT. Electrical is subject to NEC in Article 550-552. Any "travel trailer" or other RV moving on a highway will need a license plate, necessitating a safety inspection or statement of construction. You need insurance to take a vehicle on the highway and the insurer has their requirements... if you expect a travel trailer to be covered under the tow vehicle liability, read your policy.
So you are going to have huge legal entitlement issues trying to take a flimsy homemade trailer on a highway.
When the the thing gets blown into smithereens by wind forces and spreads debris all over the freeway, they'll be inclined to make an example of you and charge you with anything they can, as well as pursue you civilly for the expense of the freeway closure, and possibly impound your car in an effort to collect on same costs and fines. Insurance will be no help.
You're also likely to damage others' cars and/or hurt people. Insurance may come to your rescue there.
Structural challenges
It might be possible, with tip-top hardwoods and LOTS of bracing and elegantly placed shear walls, to get the thing to survive a gust of wind. However, a cathedral ceiling creates a very structural challenge. They're not easy to make even with proper timber. It is an inverted V-shaped form with tremendous forces trying to spread it. 1x2 lumber is simply not up to the task.
Can I build a 30-ft long, 8.5-ft wide tiny home with a cathedral roof that I can tow on highways with 1 x 2's and plywood?
Not even close.
What materials are sufficient?
Aluminum, using aircraft fabrication techniques. I would probably get a Shopbot or other CNC router intended for wood, and cut and drill the aluminum sheet on that. You'd also need some jigs to bend it.
What materials are ideal?
Stainless would be nice, since stainless does not have a fatigue limit unlike aluminum. Probably too tough to cut on a ShopBot though.
What engineering manuals and software are available for modeling dynamics?
With random dimensional lumber, it's a lost cause since the material isn't consistent enough. With the metals I mentioned, aircraft techniques will suffice.
