Can the factory card 'move' action cause combat?
Yes, in Scythe every move order can result in a combat and encounters: it's clearly stated in the manual, by the fact that there is no text at all saying that the move from the factory must be considered different from any other move.


  The rules also suggest that, in order to improve game flow, that as a
  player is taking their bottom row action, the next player can start
  taking their top row action.  


"Suggest", "in order to improve" is way different from "state that it's mandatory to". Feel like it's not working for you? Don't do it.


  However, with two move actions overlapping, it would create a great deal of confusion. 


Then avoid overlapping the two moves. If you get confused by a simple suggestion, then you need to know the game better. To be able to improve the flow of the game in respect no normal playing, you first need to know how to play the game normally -this applies to Scythe the same way it applies to Yatzhee the same it applies to Bunny Kingdom, where after a while you've played it you realise you can skip an entire turn. How many time have you played it already?


  Just how should the 'overlapping turns' dynamic work?


The "overlapping turns" dynamic works exactly as stated in the rules:


  in order to improve game flow, that as a player is taking their bottom row action, the next player can start taking their top row action. 


So, while a player is taking their bottom line action, the next player can begin taking their top row action. If it creates any kind of confusion, don't do it. If the next player thinks he wants to wait before taking his top row action, he can wait. And if the current player and the next player fail to realize something so obvious as the consequence of doing two overlapping moves and end up in a mess, then they need to play the game a few times more. Or maybe trying to play something simpler.