A: Let's be clear. You are not on the mortgage but have voluntarily paid your daughter's mortgage since 2008. Your question is if your daughter elects to sell her house, can you ask for a share of the proceeds.
The answer is no, and yes. You are a stranger to the mortgage company - you are not on the loan. You have no legal standing to ask for a share of the proceeds at closing. Between the mortgage lender and the title company, you don't exist.
Can you ask your daughter for a share of the proceeds? Absolutely. Is she legally obligated to give you a dime of the proceeds? Unless you have an agreement that she will repay you (unlikely), she is not legally obligated to give you anything.
A: Knowing how to evict a tenant is Landlord 101. Here's what I recommend that you do:
1. Hire an attorney to handle this tenant removal from start to finish. We can't get you up to speed in a Q&A forum.
2. Assuming the attorney you hire is someone whom you will want to use in the future, do that. When you have a question to which you cannot find the answer on your own, ask that attorney. You might have to pay for that sort of service but it is more than well worth the $$ spent. Hire him or her on the next tenant removal.
3. Educate yourself by reading chapter 83, Florida Statutes. Keep a copy of the documents form this eviction case for reference. Join landlord groups online or in person. This is a business, just like owning an auto repair shop or owning a trucking company. You have to know your business and how to run it. If you treat landlording as a side business, you are going to suffer the consequences of bad decisionmaking because you would be foundering around in the dark. We don't want that to happen to you. We want you to know how to pick good tenants, to have a good and enforceable lease (this is where having an attorney is critical), and how to terminate the lease properly so you are not exposed to liability.
4. As to this situation - DO NOT under any circumstances, change the locks, cut off water, electricity, or heat, or internet, or anything else. Do not hang around the property, contact the tenant incessantly, etc. Good luck!