Temporal Participial Phrases

Solution 1:

They are all adjuncts ('adverbials' in traditional grammar) in clause structure - the first two are depictive giving descriptive information, and the last one resultative expressing a subsequent situation (CaGEL p1224).

In the first two the situations are simultaneous and could be paraphrased crudely as:

He was lying on the ground and staring into the sky at the same time.

He was coming toward me and running at the same time.

In the last one the gerund-participial (-ing clause) expresses a situation subsequent to and dependent on that of the main clause, and could be paraphrased as:

When he arrived, he found no one there.