You can indeed, but it is rather literary - few people would use that construction (an absolute construction) in ordinary speech.
The second is very unlikely, for several reasons. First, the literary tone of the absolute construction doesn't match the colloquial "won't". Secondly, the construction is usually used only in narrative, so it is unlikely to be used in the present (unless it is a present tense narrative). Thirdly, especially with a "not" in the absolute clause, I would expect the main clause to refer to it in some way. Eg "He not coming, I went on my own". Note also that "He not coming" would mean something like "Because he did not come (i.e. arrive)": it doesn't mean "Because he was not coming (i.e. accompanying me)".