Mantis Eggs

I agree with John Stanard's comments, but I'll add that she can still lay eggs even if she hasn't mated. A few years ago I had a female Mantis religiosa I caught as a nymph, which molted to adulthood after a few weeks and then eventually laid an ootheca-- without ever so much as seeing a male! At least one egg hatched the next year, so there's some possibility of viable offspring via parthenogenesis. (The baby died a few days later, but that might have just been due to my poor husbandry. I need to do some follow-up experiments.)
Unless you want them to mate, why would you put the male and female together?

Do you know what species of mantis you have? How did you come into possession of her? Knowing this will help answer some of your questions. The females do not need to mate in order to become gravid. As to fertility, yours may have mated prior to coming into your possession.

I would keep the male away from her, as she may eat him if she's hungry.

I would put the oothecae outside. If you intend to keep them, I'd put them in a terrarium in an unheated garage or shed. This way you eliminated the risk of a premature emergence over the winter.