So have you checked the alternator with a VDM at the battery terminals in the engine bay? If so, what was your output reading? A regulator issue can be noticeable at the connection. I had an intermittent problem on a car that showed the alternator was good, even under a tested load, even though I took it to two different places? but in fact I new the battery was new, and that the alternator voltage reading said other wise and an intermittent problem, so I changed it. No issues.
Have you checked all possible connection.. Battery and other else where? Damaged the ECU from jump start?
The white smoke maybe an Oil Separator issue. White smoke is a common problem if the unit is failing. I use the BMW synthetic, or Mobile one. Either way, each Qt. is about the same price.