However, the more I consider the timeline and where we are at in the process, the more I WANT to close on the property in cash.I have been told by several lenders that I will then be able to refinance into an FHA 203k loan so long as I have a notarized Promissory Note that  I will be cashing-out.

Where I am if I get a signed and notarized DIL directly from the borrower, title won't insure it without a notarized affidavit that they prepare and supervise.  

Without asking me, the PM said it is okay, so the tenant moved forward with leaving her daughter the keys to the home and wrote a notarized letter stating she is leaving her daughter in charge of her responsibility and possessions in and of the house.The PM told me the neighbor daughter was just some neighbor living across the street, failed to mention the agreement he had with the daughter and the fact that she is actually the tenant's daughter and asked me to go get the keys from her when I arrive at the home.I got angry due to the amount of liability risk involved with this and told the manager he has 24 hours to fix the situation before I send him a lawyer's letter. 

There will be a notarized occupancy affidavit in the loan package stating the home will be non-owner occupied.

I imagine a quit-claim deed although I don't actually know that to be true.Which brings me back to my original point - if she is alive, which he says she is, then she would have to physically sign a quit-claim deed - with a notarized signature before he could sell the house.  

Some lenders are actually requiring the buyer's, seller's, Realtors and the title company to execute a notarized arms length to stipulate that the transaction is arms length, that the parties are not aware of any other agreements between any members of the transaction and that they will not resell for a specific time period. 3rd: The title company's underwriter.

Someone mentioned creating a notarized doc stating that after 30 days If the rent isnt paid I‘m allowed to toss their belongings out and kick them out...both of us sign and notarize it...can they still squat?

You can also followed up with a notarized letter and keep for your records.

Should we just get a notarized affidavit of small succession(I'm buying for $6,500), or do we have to have a judgement of possession in court?

Once they haven't complained for 6 months offer to Pay each of them for their time to sign over a notarized deed to you.