No, no part of these mountains is a temperate rainforest.

Um, yes there is, and no offense but why would you even reply to something you didn't know anything about? There are several small temperate rainforests throughout the Appalachians from New England to Northern Georgia. They are slowly deteriorating because of minimal awareness and construction (roads, parks, etc.).

What source of information are you baseing your awnswer on ? It is true that according to ceartain definitions of a temporate rainforest,the Appalachian region does not feature such a forest.However,most experts do classify parts of Appalachia as having a climate which supports the growth and maintainance of a temperate rainforest.
There are in fact parts of the southern Appalachians which average over 80 inches of rain per year.In NC.and Tenn.nearly every location exceding 3,500 feet of elevation recieves at least 70 inches of rain a year,while locations along the southern escarpment,as low as 2,500 feet measure that much and more,particularly the slopes facing the Atlantic and Gulf of Mexico.
In the central and northern Apps.,elevations above 4000 feet tend to recieve enough precipitation to be classified as temporate rainforest,however cooler temperatures in these locales may afford these locations a boreal type classification.To be honest,I am not ceartain as to the excat climate/forest classification of the Apps.,north of the southern mountains.
You can find a great deal of information on this subject by simply going to the NWS online climate page,and ,or The University of Oregons online interactive climate tool.Just google PRISM EXPLORER-UNIVERSITY OF OREGON.

Yes, there is. The location? Far eastern TN, and the locals there aren't the most friendly, western NC, and extending slightly into Georgia.

There's a small area of wetland... or really not even that, just a really boggy area called Stoney Creek in East TN.

You can't really get to them easily but to see them in person from the mountains is breathtaking. They appear massive and lush but actually aren't really that big compared to India's or anywhere else.

Luckily, as small as the pockets are, they're completely underdeveloped. In the middle of national parks or mountainous expanses of government property. Perhaps with good reason.

I've lived in the foothills of them, and seen the larger forests a couple times. They are in the middle of nowhere though.

Also keep in mind much of the more easily accessed state park areas are all second growth forest. Much of these forests that weren't protected by rugged mountains were cut down less then 200 years ago by the logging industry that sprouted then died. They still form canopy, albeit low ones, and a lot of ground vegetation has been introduced. We'll just have to see if it continues to grow.

And as far as location goes, keep in mind there are many small valleys and steep hollers that are tiny, no bigger then a good size estate that will receive pouring rain, while you can walk out of into the flat areas of land, and only a small drizzle of rain is happening. There's a lot of 'hills' nor more then maybe 500 ft. high that create a very rollypolly landscape.