Old question...but very common.
Yes, any one person can sell you a home at any price of their choosing.
Gift Equity is perfectly acceptable with a gift letter.
To ascertain the value, the lender will request a new appraisal.
The actual value of the home for loan to value purposes is going to be the new purchase price regardless of appraised value.
In this case $440/$550 would be 80%.
It can be one and I am guessing ....has already been done with much success!!
The new property tax bill will be calculated on your purchase price of the home.