More neighbor issues

I saw on a particular Facebook page that the neighbor downstairs has been telling people that someone - and the inference is that it was me - recently broke into her apartment late, late at night by going through the basement door into her dining room. That's just not possible because there is a door on the side of our house that leads both to the basement and to a door into her dining room that is kept locked at all times. I have the key to the basement door, but not to the door that goes into her dining room, but I would never consider going down the basement in the wee hours of the night unless I absolutely had to, like a weather emergency or something, and it being late autumn leading into winter, we just haven't had that kind of situation since sometime in the spring when the previous tenant was still living there. I have gone down the basement several times lately and it has been during the day and never did I go into her apartment. NEVER. I do not have the key to the door that goes there anyway, so I could not even get into her unit. She has made the accusation that this break in has happened several times, and I have gone down the basement several times recently to retrieve things that I have stored down there, which is my right. Apparently, she is under some kind of illusion that I am breaking into our house when in point of fact, I have every right to be able to go down the basement since my property is stored there. Now I fear that she's probably gone ahead and in her paranoia, changed all of the locks, which means that I can never again go down the basement to retrieve belongings that I have there and I won't be able to retreat there in the event of a weather emergency. 

I am at my wit's end with this neighbor and I do not know how to address this situation. She will not talk to me, won't pay her half of utility bills and I am hearing tell that she has no money, leaving me once again on the hook for paying her half of utility bills in addition to my own. She did not respond to my request for half of the most recent electric bill and shut off my heat in retaliation for it for two days until I complained to the landlord about freezing in my home. I did go out and buy a space heater, but using it has run up the electric bill by a considerable amount, so I am only going to use it if it gets really cold in my place. I have one more storm window to install and I gave it to a friend earlier this year to repair when I accidentally broke it, but I have repeatedly called him to see if he fixed it and he's not ever responded. So now I suppose I will have to measure the window to have a new one made for it. More money to spend that, fortunately, I can deduct from my rent. Anyway, these neighbor issues are really beginning to wear on my nerves and I am frustrated and have no idea of what to do or how to handle someone who accuses me of breaking into the house when I want to go down the basement using the side door on the outside of the house. It's really maddening to feel like she may even call the police if she hears me going down there again and thinks that the house is being broken into when it's just me going down there to retrieve something that I want that is in storage there. I wish I could somehow explain to her that, look, I have property in the basement that I need to have access to, and if you hear me coming into the house through the side door and going down the basement, it's just me going to retrieve something of mine. No, I am NOT breaking into the house, because I live here. But I guess that there's no reasoning with her and I will just have to be sure never EVER go down the basement again while she is home, lest she become paranoid and start making more accusations that I am breaking into my own home. I'm really beginning to regret ever asking her to move in. I thought that it would work out really well, but it's turned into a major headache and a nightmare for me and the sooner that she moves out, the happier that I will be. Lesson learned: friends do not always make the best tenants. I'm back to being stuck paying for both hers and my half of the electric and water bills unless I can somehow convince her to please pay her half by leaving a note in her mailbox asking her to please remit to me what she owes in the way of her half of utilities by leaving a check in my mailbox. That is what I did the last time and got no response. Oh, well. More neighbor drama, just what I wanted..................