Firstly, you will need a large terrarium for even dwarf trees, and secondly, even dwarf trees will outgrow a large terrarium given enough time. If your family members are bonsai experts, this shouldn't be an issue. Also note that most trees native to Oregon are accustomed to a colder period during winter, often dropping below freezing. Without this treatment the trees will not develop properly, and may not survive.
You may be able to use small ficus trees as deciduous, and Norfolk Island Pines for evergreen, but the look wouldn't be the same as your inspirational photos. Both of those trees will benefit from the humidity and heat in a terrarium. But then, they also will outgrow it given time.
The easiest thing will be the moss. Moss doesn't have a root system, so transplanting is easy, if you keep it shady and moist. Growing moss on the trees trunks will be difficult unless the trees already have corky bark.