Does anyone have any advice on how to deal with possible water damage in a rental home?

I’m renting a house that shows some signs of water damage on the kitchen ceiling. We have always rented apartments, so this is our first experience with a house. The lease is already signed and we are moving in this weekend. We mentioned the stain on the ceiling thinking it was possibly from the gas stove at first, but now I am certain it is water damage. The company we are renting from says they won’t do anything about it because it was that way when the homeowner bought the house. If the stains are old, is it something we should worry about? How can we make sure they aren’t an issue? We can’t afford to pay someone ourselves to investigate it right now. Would they be required to disclose information about that if a former inspection showed signs of mold? Or would they be responsible for covering the cost of having someone check it out? I have two small kids and am a major worrier. Just want to do what we can to make sure it’s safe. The previous tenants still lived there when we first toured the house, so we didn’t check on everything as thoroughly as we should have. We’re new to all of this and learning that we should have inspected better before signing the lease.