1. The mountains *sharp peaks pointed to the sky*. - can't really use any of the words you suggested here because your words suggest movement - but the mountain doesn't move. Even in a creative / artistic way, the idea sounds a bit strange for a mountain to raise sharp peaks.
2. Stop looking at the ground; *lift* your head up. - raise already means in the 'up' direction so including 'up' in this sentence is unnecessary if you use 'raise'
3. When the fog *lifted*, we could see the road clearly. - here 'the fog lifted' means it rose and disappeared, this is a less common meaning of 'lift'
4. The dog *rose* its head and looked at me.