Good morning, Sally,
You have a lot of questions, and I will try and sort them out and answer them for you. But if I misread any of these, or need more information to answer them fully, please get back to me and we'll go a little further.
First, whenever there are extenuating circumstances or compensating factors, understand that a human underwriter will most likely be evaluating you and making the decision. It's up to you to convince this person of the validity of your factors. Your loan officer or broker should be able to help you with this -- if your lending professional is not capable of helping you write a decent letter of explanation, find another one.
If you have more than the minimum down payment, that will of course be helpful. You did not indicate how much you can put down, and that makes a great deal of difference. It would also be helpful to know your FICO score, since you have it.
What are your extenuating circumstances? To successfully navigate this, you need to state, in easily understandable terms, what occurred to cause your credit problem (it helps if you can show that prior to the problem, you had a good credit history).
You can show on your credit report where your payment history went from good to bad, and state the nature of the problem, and why you had no control over it. For example, "Up until June of 2013, I had no late payments. Then, I was diagnosed with ______, and I was able to work fewer hours, while at the same time, my medical bills increased."
Next, you need to show the underwriter why the problem is unlikely to recur. "I have recovered and am back to working full-time, and have paid my bills on time for the last six months."
That's how extenuating circumstances work. Now, we'll tackle compensating factors. You can read about what constitutes a compensating factor here. https://themortgagereports.com/30500/what-is-the-debt-to-income-ratio-for-fha-home-loans
Compensating factors include a house payment that won't be much more than what you're paying now, enough savings (reserves) to cover several months of payments if you experience an income interruption, a higher down payment, conservative use of credit, a high FICO score, buying an energy-efficient home, and others.
Understand also that meeting official FHA guidelines may not be enough for many lenders. They apply what are called "overlays," tougher-than-required guidelines. They are allowed to do this, so if you need a lender that follows FHA guidelines to the letter, you'll need to ask lenders you contact if they apply overlays, or what their minimum FICO is for a loan, or similar questions.
Thank you for writing and good luck.