My husband and his sister inherited a house from his parents a year ago. My husband and I have been living in the residence and paying the property taxes and home owners insurance as well as all the other living expenses. We have also put money into remodelling the home in order to bring the home to liveable conditions. This has been out of pocket expenses for my husband and I. His sister has paid no out of pocket expense for the maintaining of the home, taxes or anything. She is requesting us to pay her rent for living here. Is it legal for a co-owner to request rent from another co-owner on an inheritance property? Also the expenses that we have put into the house--would those be just considered living expenses or do those get reimbursed by the sale of the home or split in cost with his sister?

You should be paying 1/2 the appropriate rent in my opinion. If another party were renting the place, you would be entitled to half the rent.
Also, improvements to the residence should be reimbursed to you by the estate provided the improvements are approved in writing by the sister.
I have no expertise in this field. My opinions are based on common sense only.