I would probably look into a hydraulic piston for automotive applications. If you own a hatchback, van or SUV with a flip rear or any number of other autos with a door that opens up. One should be sufficient to overcome the weight of the monitor unless you went crazy on the size.
For the latch, I would go magnetic and use the type that "lock/unlock" into position. Super common on entertainment centers though kitchen cabinets use them except they don't have a lot of travel. However IKEA uses magnetic latches that have nearly an inch of travel.
I would opt for this method because of the use of standard parts, hence, a little easier to source.
I can't see the Project Morph videos so my apologies if that's how it was done.