raise - lift - pick up
1. The mountains ..... their sharp peaks in the air. (I think - raise)
2. Stop looking at the ground; ..... your head up. (I think - raise)
3. When the fog ....., we could see the road clearly. (I think - raise)
4. The dog ..... its head and looked at me. (I think - raise)

Question

raise - lift - pick up
1. The mountains ..... their sharp peaks in the air. (I think - raise)
2. Stop looking at the ground; ..... your head up. (I think - raise)
3. When the fog ....., we could see the road clearly. (I think - raise)
4. The dog ..... its head and looked at me.  (I think - raise)

Answer

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.

Answer

Hi Sergey,
1. The mountains ..... their sharp peaks in the air. This is very descriptive and difficult to know what the writer would say.
2. Stop looking at the ground: lift your head up. 
3. When the fog lifts, we could see the road clearly. 
4. The dog raised its head and looked at me.

Hope that helps a little :)