Complicated FHA Loan and Divorce

[Shane Hyde]

First post! I have been lurking for a few months and feel like I am starting to understand enough to ask specific enough questions, so here goes!  (Note: I will update my profile and do a noob post this week)

I recently finalized a divorce and the remaining asset between us is the house we bought 6 years ago. It's worth $400 and has $308k remaining on the loan. She wants to keep it, so I agreed to take $30k on a buyout if she can secure funding, if not we sell and split whatever is left 50/50. As you can see by the gap, I'd prefer she buy me out. 

Only my name is on the mortgage, but her name in on the deed.

Question: If she were able to secure funding, would this be a purchase or refi since her name is on the deed?

Question: Can existing equity be used for her down payment? - I was hoping that this could allow an FHA 3.5% for her leaving plenty of equity ($92K) to buy me out and cover the down payment and fees ($30k to me, $14k down, $3k fees). If this works she would only be at $355k.

Heart of the Question: Do traditional lenders REQUIRE actual cash, or can equity in home count?

I'm really getting excited to get into at least a house hack in the next few months, even though I live in MA, and clearing this up will really help me move forward. I've attended two meetups and really starting to get the idea

Thank you to anyone who sees fit to chime in!

Shane