I can arrange to get you our Private Funding.

He may be able to arrange for a bridge loan so you can close on time and become an equity partner. 

I was suggested to keep the suggested that I should make an arrangement to buy the house at the fair price, but keep it in her name, dont move the loan.

As long as it isn't close-to, or more than, the amount you'd pay for other living arrangements, it can be a good deal.That being said, you'll want to run two analyses for a house-hack.

They were fine with this arrangement, and they paid rent in 10/2018, 11/2018, 12/2018.  

Does anyone have any experience with this arrangement?

The REALTOR is required by law to provide proper notice and they can't require you to move furniture or arrange your belongings a certain way.

I made arrangements  with my family to buy the house.

If you're using the money from a HML to do a fix and flip, it is possible to arrange a deal where you pay them back after you sell the house and close escrow. 

Hours spent by any person with respect to the owner’s capacity as an investor, such as arranging financing, procuring property, reviewing financial statements or reports on operations, planning, managing, or constructing long-term capital improvements, and traveling to and from the real estate are not considered to be hours of service with respect to the enterprise.