Ddot did a good job of summarizing your situation. I'll try and help out a little bit as I work in the mortgage industry.
If you have the credit scores to qualify for a conventional mortgage I would definitely go that route. Assuming you live in Nebraska (based off of your sig) you could get a 30 yr conventional/conforming mortgage ranging from 4.875-5% (based off of todays Wells Fargo rate sheet). By putting that much money down you would be looking at a note of roughly $80,000, a Principal and Interest payment ranging from $423.36-$429.45 then you would have to add on taxes and insurance which probably won't be more than $150-200/month so a potential mortgage payment of $580 (which is very good all things considered). I am not too sure about how closing costs or down payment gifting goes for a conventional mortgage, but I think you can get a gift letter that can be accepted by the lender.
Additionally the most important aspect of you borrowing the money is your debt to income ration. Lenders will pull your credit report and add up all of your debts (CC payments, auto loans, installment debts, student loans*, any mortgages you have, etc). They generally won't approve your loan application unless your back end debt ration is under 50% meaning your debt obligations can't be more than half of what your average monthly income is. * In the case of student loans if you have deferred payments, you must prove that they will be deferred for the next 12 months to have them not count in your DTI ratios.
If you were to go through the FHA, your rate could be lower by 1/8 of a point given todays rates (they fluctuate daily), but you would have to pay an upfront mortgage insurance premium (UFMIP) that is currently 1.75% of your loan amount and is increasing to 2.25% at some point in April. You also have monthly mortgage insurance added to your escrow account which .005-.0055% of the final loan amount each month for the first 5 years of your mortgage.
Majority of loans today do not have a pre-payment penalty so you don't have to worry about that. The only thing you really have to worry about is you DTI ratio. Even if it is too low you could have a relative co-sign as a non-occupant co-borrower, which essentially allows you to qualify with their income, but they would be on the loan and any late payment would have an adverse affect on their credit.
EDIT- To answer your question, the remainder of the mortgage note will have a payoff that includes whatever principal balance you have and one months interest, because interest is payed in arrears.
