Or not keeping the property in a livable condition.

Should I close, if yes with what conditions ?

One approach could be to let her know you will let her use the last month's rent as her deposit if she makes sure the house is in good condition and clean when she moves out. 

@Khayla WilliamsI strongly recommend against borrowing 100% for a propertyThere is significant risk in owning real estate including section 8 and maintaining the home in a certain condition and if you do t have the funds for those repairs you could lose it all.I always recommend people budget and save enough for a down payment and reserves before getting into their first property.

However, a quick glance at Zillow reveals some new or decent condition rental listings lingering for over 30 days.

Are there addition conditions I should consider if I do lease to them?

You must use numbers that are representative of the property’s condition if you are not going to renovate it before putting it in service.

That's all I care weather anyone else likes or not.

I've focused my attention in the area between Hwy 44 and Gravois Ave from Tower Grove Park down to River Des Peres for a nice intersection of price and condition.

I hesitate to use the term "slum lord" but the conditions some of these residents are living with are not healthy.