I have heard that the Appalachian Mountains is qualified as a temperate rainforest. What qualifies a temperate rainforest and what makes the Appalachian Mountains a temperate rainforest?
The definition of temperate rainforest is rather vague and varies around the world, but in general they are forests well away from the tropics which have a mild, damp climate, and prolific rainfall. Trees can be coniferous, evergreen broadleaf or deciduous broadleaf. Welsh rainforest (I prefer to call it woodland) is mainly deciduous oak, and rather bleak in winter. I have no experience of Appalachia, but I do have experience of rainforest in the west of Britain and rainforest in Malaya and Borneo, so I'm in a good position to make a comparison.
Anyone hoping to see rainforest in Britain which bears any resemblance to the rainforests of Malaysia will surely be disappointed, though in some other parts of the world there are temperate rainforests with a sub-tropical flavour. This is not to say that in UK rainforests there is nothing worth seeing. There is quite a lot, particularly in summer, and the same is probably true of Appalachian rainforests. Our UK rain forests are particularly rich in lichens, for those with an interest in lichens. Unfortunately some British rainforests are in danger of being taken over by rhododendrons, which are an invasive species and not native. Tropical rainforests are incomparably richer in species, plants as well as animals, but have the disadvantage that they are also far more dangerous.
Yes, the Appalachian Region would be considered a Temperate Rainforest based on a combination of rainfall numbers, humidity and temperatures. Its latitude runs from in the 30's and up to high 50's in latitude location north of the equator. Feeding the forest region are a combination of two damp air masses which rotate in a cyclonic rotation inward from the marine locations, namely the Maritime Tropical air mass bringing very humid air from the Gulf of Mexico and the cooler air but also damp Martime Polar air mass. The mountains, while lower in elevation than the Western Rocky Mountains, offer much of this damp air from either air mass to move over toward the mountains when the jetstream is positioned in a way to not influence these air masses in a way that pushes them eastward. Orographic type precipitation can occur as a result, but much of it in that region is frontal as the colder Contintental Polar and Arctic air masses tend to interact with the damp air masses mentioned previously dumping significant amounts of rainfall suddenly as I can attest in my many journeys through these mountains on my way to Georgia or Florida from Ontario. Given the rainfall alone and the fact there is a diverse complexity of vegetation and animal specie, I would surely believe that the Appalachian Region qualifies easily as a Temperate Rainforest. Climate graph > location in Appalachian Region https://en.climate-data.org/north-america/united-states-of-america/virginia/appalachia-135846/#climate-table and, location map of North American air masses paths running along and to this region https://wordpress.accuweather.com/wp-content/uploads/2018/01/static-winter-air-masses-11-am-1.jpg?w=632