Answer

As for placing it on a fire:
For cold food, it is forbidden to place it on the fire even when it is fully cooked, so certainly that would be true when it has only reached the Ben Drusai stage. If it is hot and one wants to place it in a hotter spot, it seems that even when it is hot this is forbidden, for even leaving it on the fire from before the Sabbath is forbidden when the cooking improves the dish (although Sabbath 20a implies that it is permitted). Still, the practice is to be lenient about leaving it on the fire in pressing circumstances.
Here I found an overview of the laws of cooking according to the basic law: https://www.hidush.co.il/hidush.asp?id=9774
As for placing it on a hotplate:
It seems to me that this is forbidden, since this involves cooking at least on a rabbinic level, and what was permitted on a hotplate is reheating, not cooking.