How to structure buying a property with mortgage + family loan?

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Hi all,

Would love a few different perspectives/advice on my situation.

My grandparents passed away and left my mom, her sister, and her brother a fully paid off property recently appraised for $850,000. They would like to keep it in the family and have agreed to sell it to me for $750,000. 

I've been pre-approved for a $500,000 mortgage, meaning I could buy 2/3rds of the home. My mom has agreed to keep her 1/3rd interest in the property until I can pay her off, which would realistically be in 7-10 years.

What's the best way to structure this deal? Does my mom keep her 1/3rd interest and I apply for a $500,000 mortgage to use as my primary residence, while keeping her on title? Or is it best if she "gifts" me her 1/3rd portion and I can make a side agreement with her to pay her off over time?

For tax purposes, what's the best way to work the deal with my mom? She wants $850 per month "rent" in exchange for her 1/3rd ownership portion, which will decrease accordingly as I pay off her 1/3rd interest. 

Also, I hope this doesn't come off as insensitive, but I will be doing extensive remodeling to this property. Once done, I expect it to be worth around $1,150,000-$1,200,000. While my mom owns her 1/3rd interest, I'm worried that I will be paying more than I can afford for her 1/3rd due to the work I've put into to the property - which otherwise, she would have to sell if I wasn't buying the home. How is a fair way to justify a future purchase price, given this home is located in an area that is becoming heavily gentrified? Is it a smart idea to agree to a price NOW with a balloon payment due in X years? How does this typically work?

Any advice would be great appreciated - thanks!