As a lecturer, can I use the same problems used in another course in my own course?
Check if the MOOC has a license assigned to it. If it is for example Creative Commons Attribution (CC-BY) you are allowed to use the questions as long as you give a reference to the original creator. You don't need to ask the MOOC in this case. CC-BY-NC (non-commercial) might be a difficult case, because you teach this course as a business and earn money for it. 
A second option is to directly ask the MOOC creators. 
Adapting the question significantly is a third option, as long as you don't copy directly texts from the original question and just base it on the general idea this would be fine.