I don't think you can use the equity in your mother's home to apply toward your down payment (at least in my 30 plus years of experience with FHA loans and real estate).
The minimum down payment requirement is 3.5% of the purchase price. You are allowed to use either your own funds OR gift funds, but not borrowed funds. (Be careful, the site link above has old information on FHA loans from 2009 and previous so get your info from a mortgage banker instead).
The appraiser will appraise the house for current market value and the lender will use as your maximum loan amount either the appraised value or the contract price, whichever is lower.
Don't use Quicken Loans they are very, very uneven in their approval process. Their business model includes phone reps that are not familiar with underwriting so you end up blowing $500 to $700 on the appraisal and end up with no deal in many instances because they will tell you things that underwriters won't accept (like using your Mother's equity for your purchase).
Wells Fargo is moving to the same model: phone reps to take the app and no experienced loan officers.
Not too many credit unions do FHA. They generally do conventional loans only.
Get with an experienced mortgage banker in your area. You will need to get references to good mortgage bankers either through local Realtors or friends and family that have recently purchased (not refi'ed) through a local mortgage banker. They also sell your loan, but after the deal is funded usually so you have nothing to worry about. They also have many programs with which to underwrite you and are more familiar with government loans (FHA/VA).
The difference between a mortgage banker and a mortgage broker is huge: the mortgage banker funds the loan and has an underwriter that can give you a decision based on their criteria. When you get an approval from a mortgage banker, you know the deal will close because you have been underwritten by the actual lender.
A mortgage broker never funds the loan; they just sell the deal to a lender that will fund with your requirements. However, generally when the broker is shopping he is shopping his commission, not your deal. In my 30 yrs experience with hundreds and hundreds of FHA loans, the only time I've had difficulty is when a mortgage broker (that has no funding capacity) gets involved. They promise more than they can deliver. However, having said the above, there are some mortgage brokers that are worth their weight in gold. They know what they are doing - but it is increasingly rare, especially in this market. They are not trained as well, have no onsite underwriters and in general don't have the resources that mortgage bankers have for you. Be very, very careful when dealing with a mortgage broker.
Your loan officer is key to having a smooth purchase and funding of that purchase. Don't under estimate the importance of the loan officer.
