Yes to the first. He continues to bat in the same spot in the batting order and the pitcher now must bat in the position that the third baseman had batted in (assuming he’s removed from the game and not moved elsewhere – but somebody on defense has to leave and that batting order spot now goes to the pitcher.)
The answer to the second is no I think
OldGuy is totally right. In Case 1, the new pitcher must bat for himself in the position previously occupied by the third baseman Chapman is replacing (or some other position if you switch out more than one defensive player. For instance, suppose your batting order is:
It’s the ninth and you want Chapman at third because you’re holding a tight lead, so Jordan Romano enters as pitcher, Chapman is playing third, and Biggio is out, but Romano must now bat for Biggio:
Romano could of course bat elsewhere if you make a double switch at two positions, so suppose you want to further beef up your defense by replacing Hernandez in right field with Ramiel Tapia. Because you are replacing both Biggio and Hernandez you can move Romano to fourth, exchanging the lineup spots occupied by the right fielder and DH. (If the 8 spot is coming up next in the bottom of the ninth this would make sense to do.)
Case 2 is not permitted. The central underlying rule of the DH spot is that the Designated Hitter’s position in the batting order is absolutely fixed. There is nothing you can ever do that would change the place in the batting order occupied by the DH. As we have started here with the DH batting sixth, the DH must always bat sixth so moving Giancarlo Stanton from LF to DH would mean the DH must have moved in the batting order; if Stanton the LF is batting cleanup, well, the DH couldn’t have been batting cleanup, so you would be moving the position of the DH, which can never happen.
Thanks, that all makes sense. I guess I never paid much attention to the rules governing the DH. I’m going to bookmark this excellent post for when I’ve forgotten it all again.
Not that I’m saying that would never happen, but I imagine that it’s a rare thing that the guy you have DHing would also represent a significant upgrade at a defensive position, particularly one like third base. 
What could be more likely is that your DH is your backup third baseman (for example), and the starter, who is a much better fielder than the backup/DH, has to leave the game due to an injury. The manager would face the problem of either (a) putting his backup 3B in the game, but losing access to having a DH, or (b) having an even worse player finish the game at third.
It’s not the norm, but it’s not rare - the DH is regularly used to give regular position players a day off in the field so the manager can keep their bat in the lineup, to give the rookie some experience, or it’s a platoon situation.
I think more common is to use one catcher to DH so if the other catcher gets hurt or ejected, the DH catches and the DH is lost for the game. Or a game goes quite long you could run out of pinch hitters and the DH is lost that way. Of course with the peewee league runner at second rule, that will be less often.