I would suggest: He's singing while he runs. He's drinking coffee while he walks.

They are very unnatural. He’s running and singing. He’s singing while running. (For both, adding ‘a song’ adds no information and would be skipped) He’s walking and drinking coffee. He’s drinking coffee while walking.

Those are good sentences, though they might be improved by inserting commas. I will talk about #1. "Running" is an adjective and "singing a song" is an adjective phrase. Both of these adjectives modify "he", the subject of the sentence. "Running" and "singing" are not verbs. The only verb in the sentences is "is". The adjectives can be placed anywhere and in any order: (1) Singing a song, he is running. (2) Running, he is singing a song. (3) He, singing a song, is running. (4) He, running, is singing a song. (5) He is singing a song, running. (6) He is running, singing a song. All of these sentences have the same meaning. The sentences might be made more clear by inserting a conjunction but it is not necessary to do so.