Things which can be in a mountain are principally those which intrude inwardly upon its profile or its expected profile, or are entirely encompassed by its expected profile. For example, there can be a hole, or a ravine, or door in a mountain, -- as there can be a hole in an apple, -- because they are inward projections or modifications when compared to the expected profile of the mountain. "The door in the mountain leads to a magical kingdom".
There can be gold, or other minerals, in a mountain, -- as there can be a worm in an apple, -- as they are encompassed by its profile. But these are also inside the mountain (or apple), which includes only those things which are in a mountain and which are also not evident from its surface. In these contexts inside is more likely to be used, but in is also correct. "There is a dragon inside the mountain".
More generally, surface features of any kind are on the mountain (including those things which are in the mountain but not inside it, such as ravines). This includes boulders, ravines, and tea shops. "There is a tea shop on Mount Snowdon".
Something as fundamental to a mountain as a peak, however is of the mountain, as it is a fundamental constituent part. In contexts where, for example, a dragon is known -- distinctively and crucially -- to sleep inside Mount Fire, it can be referred to as the dragon of Mount Fire as it is an important constituent of that mountain. "The north face of the Eiger is hellish".
Prepositions are slippery and complex in all language. Good luck.