@Cathy L.
Wow! You are in a tough spot and wish you the very best!
I'm not going to beat you up on tenant screening as I see that your are new to BP so first off WELCOME!
We use a 17-page lease appropriate for the state we lease in and it has been reviewed by a lawyer and then on top of that we add an addendum for pets, an addendum for moving out procedures, etc. I realize that info would have been helpful a while ago but nevertheless, I think it'll be helpful for your next "lined up tenant." BP has a FANTASTIC tenant screening product that I highly recommend. My advice on it: Don't deviate as it will cost you later in headache and money.
What to do with the tenant in the unit that is giving you the current headache, it is tough to say as we don't know what type of contract you have set up with them or what it says. I would highly encourage you to find a RE Lawyer that will look at your current lease with the current tenant along with their "demands" as realistically, paying a RE lawyer a couple of hundred bucks is going be a worthwhile education, plus, may save you thousands on that weird agreement the tenant is "demanding" you sign.
Being a landlord is very tough and I believe it to require STRONG LEADERSHIP skills. Don't be bullied by a bad tenant! They have probably done this to others and have developed the knowledge and experience base.... and.... they will do it to another landlord! I'm sorry to hear about your situation and wish you the best. Get ARMED with information and under no circumstances, get into an argument or verbal-shouting match (I know it's tempting) with them as they can become emotional and you have to walk a very DELICATE line when dealing with problem tenants like this.
I'm NOT providing legal advice: but if you have a lease (contract law) and they are in default, it doesn't take a genius to decipher who is in the wrong. Keep us updated and again, good luck!