That would be dictated by what the locals say habitually. If, for example, you are writing fiction and you get to choose, it could go either way

It it dependent upon what the locals say and what is says on maps of the area. Both are perfectly fine. A mountain is a mountain that can only be called a mountain if it reaches a minimum height. A mount is a much smaller mountain or a bump in the ground from which in times gone by before cities were built you could get a view of the local country-side. They are common places names on all English maps and maps in English speaking countries. A good example is 'Mount Pleasant' in London a major post office sorting office. the "Mount" has gone now, and there are many places in England called "Mount Pleasant" there is not just the one famous "Mount Pleasant" sorting office.

Seeing from your answer that it is a real mountain and that you are translating, one possibility is not to translate the name at all--just transliterate the real name. The easy way out is to see what English-language maps and atlases do. We don't translate Mont Blanc, Acongagua, Kilimanjaro, Iztaccíhuatl, Mauna Loa, etc. and we don't add the words "Mount" or "Mountain" to them. On the other hand we do talk about Mount Fuji and Mount Vesuvius, so, again... there's no rule. It should be possible to make it clear from context that it is a mountain.

There are lots of Bear Mountains in the US. Less important mountains are often named this way with a common noun followed by ‘Mountain’ ‘Root’ isn’t a natural name for a mountain though.

There's no rule. The answer is to look it up. Usually, any map, including Google Maps, will give you the answer. In the United States, these things are officially decided by the U.S. Board on Geographic Names, which usually follows local tradition. Bear Mountain is Bear Mountain, Mount Rushmore is Mount Rushmore, Pike's Peak is Pike's Peak, and Denali is just Denali. If you know the official name of the mountain, it is not correct to change the form. You mustn't call Mount Washington "Washington Mountain," and you mustn't call Steamboat Mountain "Mount Steamboat."