Are married students allowed to stay with their spouses at IIM Kozhikode? If not, then can they live outside the campus?

Married accommodation is not readily available on campus, but on request, IIMK can facilitate locating suitable accommodation for students, in proximate areas. 

My sister's batch had 2 married students. The institute assigned them an extra room for spouses in the women's hostel. If your roommates are okay with it, then your spouse can visit you for a few days. 

Outside facilities are also available for spouses' accommodation. But then the transport expenses will have to be taken care of.